[{"data":1,"prerenderedAt":1140},["ShallowReactive",2],{"layer:data-handling:understand":3},{"layer":4,"contentHash":1120,"dependencyHashes":1121,"approval":1133,"releaseId":1139},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1115,"reviewStatus":1116,"authoring":1117},1,"data-handling","en","understand","Organise, picture, summarise: how the methods work","Kinds of data, tables and graphs done properly, and exact methods for mean, median, mode and range","Tell categorical from numerical data, build self-checking frequency tables, choose a key or scale for pictographs and bar graphs, and use exact methods for mean, median (odd and even counts), mode (two modes or none) and range, even from a frequency table.",[13,14,15,16,17],"Classify data as categorical or numerical, and numerical data as counted (discrete) or measured (continuous).","Build a frequency table from raw data and use it to answer questions.","Choose a key for a pictograph and a scale for a bar graph, and draw and read both accurately.","Calculate the mean, median, mode and range of a list, including an even number of values, two modes and no mode.","Find the mean, median and mode from a frequency table, and avoid the most common mix-ups.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Tally marks, fair share (Discover)",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Data-type sort, two cities, quiz marks, mean match, sprint",{"label":38,"value":39},"Key formula","mean = sum ÷ count",[41,45,51,57,60,84,89,173,176,181,184,217,242,252,265,270,273,307,312,323,328,331,355,382,390,395,419,424,427,439,455,465,470,482,486,491,494,504,513,517,528,533,536,564,572,576,594,599,602,636,666,671,682,687,690,719,729,761,773,812,817,857,885,899,951,1072,1087,1091,1097,1101,1105],{"id":42,"type":43,"markdown":44},"intro","prose","In Discover you met data, tally marks, pictures and four friendly summary numbers. Now we slow down and make every method **exact**. What kind of data do you have, and what can you do with it? How do you pick a key so a pictograph does not need 60 little pictures? What is the median when there are two middle values? What if two values tie for the mode, or none repeats at all? And how do you find the mean of 30 children's marks without writing 30 numbers in a row?\n\nBy the end of this layer you should be able to take any small set of data, organise it, draw it and summarise it, and explain **why** each step works.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this lesson","Each chapter has a method, a worked example and a quick practice question. Try every practice question before opening the hints. The common mix-ups are collected in Chapter 10, but each chapter also warns you about its own trap.\n\nThe class surveys, quiz marks and school figures in this lesson are **invented examples**. The Delhi and Chennai temperatures are real: they are the India Meteorological Department's long-term averages for 1991–2020 at New Delhi (Safdarjung) and Chennai (Nungambakkam), rounded to whole degrees.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","What kind of data is it?","Chapter 01","1 Kinds of data",{"id":58,"type":43,"markdown":59},"kinds","Before you can summarise data, you need to know **what kind** of data it is, because that decides which tools work.\n\n**Categorical data** puts each person or thing into a **category** (a group with a name): favourite colour, mother tongue, blood group, mode of transport, state of birth. You can **count** how many are in each category, but you cannot add or average the categories themselves. \"Hindi + Tamil ÷ 2\" means nothing.\n\n**Numerical data** is made of **numbers that measure or count something**: height, marks, runs, rainfall, number of siblings. You can add them, order them and average them.\n\nNumerical data comes in two flavours:\n\n- **Discrete (counted)** data can only take separate values, usually whole numbers: number of siblings (0, 1, 2…), runs off a ball, pages in a book. You cannot have 2.5 siblings.\n- **Continuous (measured)** data can take any value in a range, limited only by how precisely you measure: height (131.4 cm), mass, time, temperature, rainfall.",{"id":61,"type":62,"caption":63,"columns":64,"rows":69},"table-kinds","table","Kinds of data at a glance",[65,66,67,68],"Kind","Examples","Can you find the mean?","Best pictures",[70,75,80],[71,72,73,74],"Categorical","Favourite snack, blood group, language, colour","No; use the mode","Pictograph, bar graph, pie chart",[76,77,78,79],"Numerical, discrete (counted)","Siblings, runs per over, goals, pages","Yes","Bar graph, dot plot",[81,82,78,83],"Numerical, continuous (measured)","Height, mass, time, temperature, rainfall","Grouped bar graph (histogram), line graph",{"id":85,"type":47,"variant":86,"title":87,"markdown":88},"pin-code","misconception","\"If it is written with digits, it is numerical\"","A PIN code (560001), a cricket jersey number (18) or a roll number (27) is written with digits, but it is really a **label**. The mean of three PIN codes is meaningless, and jersey 18 is not \"twice as much\" as jersey 9. Ask yourself: *does adding these make sense?* If not, the data is categorical, however it is written.",{"id":90,"type":91,"component":92,"componentVersion":5,"config":93,"objective":171,"textAlternative":172},"lab-kinds-sort","interactive","sort-game",{"prompt":94,"bins":95,"items":105,"seconds":170},"Sort the cards. Is each one categorical, numerical and counted, or numerical and measured?",[96,99,102],{"id":97,"label":98},"cat","Categorical (labels)",{"id":100,"label":101},"count","Numerical: counted",{"id":103,"label":104},"measure","Numerical: measured",[106,110,114,118,122,126,130,134,138,142,146,150,154,158,162,166],{"id":107,"label":108,"bin":97,"why":109},"k1","Favourite colour of each child","Colours are names of categories; you can only count them.",{"id":111,"label":112,"bin":100,"why":113},"k2","Number of siblings","You count siblings; only whole numbers are possible.",{"id":115,"label":116,"bin":103,"why":117},"k3","Height of each child in cm","Height is measured and can be 131.4 cm or 131.45 cm.",{"id":119,"label":120,"bin":97,"why":121},"k4","Blood group (A, B, AB, O)","Blood groups are categories, not amounts.",{"id":123,"label":124,"bin":100,"why":125},"k5","Runs scored off each ball","Runs are counted in whole numbers: 0, 1, 2, 3, 4, 6.",{"id":127,"label":128,"bin":103,"why":129},"k6","Time to run 100 m","Time is measured, e.g. 15.73 seconds.",{"id":131,"label":132,"bin":97,"why":133},"k7","PIN code of your home","A PIN code is a label; averaging PIN codes means nothing.",{"id":135,"label":136,"bin":103,"why":137},"k8","Daily rainfall in mm","Rainfall is measured, e.g. 12.6 mm.",{"id":139,"label":140,"bin":100,"why":141},"k9","Number of pages in a book","Pages are counted in whole numbers.",{"id":143,"label":144,"bin":97,"why":145},"k10","Mother tongue","Languages are categories.",{"id":147,"label":148,"bin":103,"why":149},"k11","Mass of a school bag","Mass is measured with a scale, e.g. 4.3 kg.",{"id":151,"label":152,"bin":100,"why":153},"k12","Cars owned by each family","You count cars: 0, 1, 2…",{"id":155,"label":156,"bin":97,"why":157},"k13","Jersey number of a cricketer","A jersey number is a name tag, not an amount.",{"id":159,"label":160,"bin":103,"why":161},"k14","Temperature at noon","Temperature is measured and can be 34.6 °C.",{"id":163,"label":164,"bin":97,"why":165},"k15","Mode of transport to school","Bus, walk, cycle are categories.",{"id":167,"label":168,"bin":100,"why":169},"k16","Goals scored in a football match","Goals are counted.",0,"Decide whether each piece of data is categorical, counted numerical (discrete) or measured numerical (continuous).","A sorting game with three bins and 16 cards.\n\n- **Categorical (labels):** favourite colour, blood group, PIN code, mother tongue, jersey number, mode of transport. These name groups; digits in a PIN code or jersey number are only labels.\n- **Numerical, counted (discrete):** number of siblings, runs off each ball, pages in a book, cars owned, goals scored. Only separate whole values are possible.\n- **Numerical, measured (continuous):** height, time to run 100 m, daily rainfall, mass of a bag, noon temperature. Any value in a range is possible, depending on how precisely you measure.\n\nTest for categorical: does adding the values make sense? Test for continuous: could a value like 12.37 make sense?",{"id":174,"type":43,"markdown":175},"primary-secondary","One more pair of words. **Primary data** is data you collect yourself, for your own question: your class survey, your rain gauge readings. **Secondary data** was collected by someone else and you reuse it: IMD rainfall records, census tables, a newspaper's cricket statistics. Secondary data saves time, but you should always ask **who collected it, when, and how**, because you cannot check the collection yourself.",{"id":177,"type":53,"title":178,"eyebrow":179,"navLabel":180},"ch02","Frequency tables that check themselves","Chapter 02","2 Frequency tables",{"id":182,"type":43,"markdown":183},"freq-method","In Discover you tallied categories. The same method works for numerical data, with one improvement: **list the values in order**, including any value that has frequency 0, so the table shows the shape of the data.\n\nClass 7A asked each of its 30 children, *\"How many brothers and sisters do you have?\"* The raw answers were collected on a sheet. Tallying them, in order from 0 upwards, gives this frequency table.",{"id":185,"type":62,"caption":186,"columns":187,"rows":192},"table-sib","Number of siblings of 30 children in Class 7A",[188,189,190,191],"Siblings","Tally","Frequency","Running total",[193,197,202,207,210,213],[194,195,196,196],"0","||||","4",[198,199,200,201],"1","卌 卌 ||","12","16",[203,204,205,206],"2","卌 ||||","9","25",[208,195,196,209],"3","29",[196,211,198,212],"|","30",[214,215,216,215],"**Total**","—","**30**",{"id":218,"type":219,"title":220,"items":221},"steps-freq","steps","Making a frequency table",[222,226,230,234,238],{"title":223,"tag":224,"text":225},"List the possible values","in order","Write every value from smallest to largest in the first column, even ones nobody chose yet.",{"title":227,"tag":228,"text":229},"Tally in one pass","cross off as you go","Go through the raw data once; make one stroke per value and cross the value off the list.",{"title":231,"tag":232,"text":233},"Count the tallies","frequency","Write each total as a number: that is its frequency.",{"title":235,"tag":236,"text":237},"Check the total","must match","The frequencies must add up to the number of observations. Here 4 + 12 + 9 + 4 + 1 = 30 ✓.",{"title":239,"tag":240,"text":241},"Add a running total","optional","Adding frequencies as you go down (4, 16, 25, 29, 30) makes the median easy to find later.",{"id":243,"type":244,"title":245,"problem":246,"steps":247},"we-freq-read","worked_example","Answering questions from the table","Use the siblings table. (a) How many children have at least 2 siblings? (b) What fraction of the class has no siblings? (c) How many siblings do all 30 children have altogether?",[248,249,250,251],"(a) \"At least 2\" means 2, 3 or 4: 9 + 4 + 1 = **14 children**.","(b) 4 children out of 30 have none: 4\u002F30 = **2\u002F15** of the class.","(c) Each row contributes (value × frequency): 0 × 4 = 0, 1 × 12 = 12, 2 × 9 = 18, 3 × 4 = 12, 4 × 1 = 4.","Total siblings = 0 + 12 + 18 + 12 + 4 = **46**. Keep this idea of value × frequency; it is how we will find the mean from a table in Chapter 9.",{"id":253,"type":254,"itemId":255,"prompt":256,"check":257,"hints":260,"feedback":262},"pr-freq","practice","data-handling.understand-freq-atmost","From the siblings table, how many children have **at most 1** sibling?",{"kind":258,"answer":259,"tolerance":170},"number",16,[261],"At most means \"this many or fewer\".",{"correct":263,"incorrect":264},"Right: \"at most 1\" means 0 or 1, so 4 + 12 = 16. The running total column shows 16 directly.","\"At most 1\" means 0 or 1 sibling. Add those two frequencies.",{"id":266,"type":53,"title":267,"eyebrow":268,"navLabel":269},"ch03","Pictographs with a well-chosen key","Chapter 03","3 Pictographs",{"id":271,"type":43,"markdown":272},"picto-key","During Van Mahotsav week, five schools in a district planted trees: 120, 90, 150, 75 and 105. To draw a pictograph, the first decision is the **key**. If 🌳 = 1 tree, School C needs 150 pictures. Absurd. If 🌳 = 100 trees, every school gets about one picture and the differences vanish.\n\nA good key: (1) keeps the largest row to a comfortable number of symbols, often under 10; and (2) divides the values neatly, or at least into halves. Here every value is a multiple of 15, and 30 divides them into whole or half symbols. So choose **🌳 = 30 trees**.",{"id":274,"type":62,"caption":275,"columns":276,"rows":281},"table-trees","Trees planted in Van Mahotsav week. Key: 🌳 = 30 trees; ◑ = half a symbol = 15 trees",[277,278,279,280],"School","Pictograph","Trees","Working",[282,287,292,297,302],[283,284,285,286],"School A","🌳🌳🌳🌳","120","120 ÷ 30 = 4 symbols",[288,289,290,291],"School B","🌳🌳🌳","90","90 ÷ 30 = 3 symbols",[293,294,295,296],"School C","🌳🌳🌳🌳🌳","150","150 ÷ 30 = 5 symbols",[298,299,300,301],"School D","🌳🌳◑","75","75 ÷ 30 = 2.5 symbols",[303,304,305,306],"School E","🌳🌳🌳◑","105","105 ÷ 30 = 3.5 symbols",{"id":308,"type":47,"variant":309,"title":310,"markdown":311},"picto-limit","model_limit","What pictographs cannot do well","Pictographs are friendly, but they are **approximate** when values do not divide neatly. With 🌳 = 30, a school that planted 100 trees needs 3⅓ symbols, and nobody can draw a third of a tree accurately. For exact comparisons, and for data with many categories or large, uneven values, a **bar graph** is better.",{"id":313,"type":254,"itemId":314,"prompt":315,"check":316,"hints":318,"feedback":320},"pr-picto","data-handling.understand-picto-key","In a pictograph of milk sold by a dairy, the key is 🥛 = 20 litres. How many symbols (including a half) are needed to show 150 litres?",{"kind":258,"answer":317,"tolerance":170},7.5,[319],"Each symbol is 20 litres.",{"correct":321,"incorrect":322},"Yes: 150 ÷ 20 = 7.5, so 7 full symbols and one half symbol.","Divide the value by the key: 150 ÷ 20.",{"id":324,"type":53,"title":325,"eyebrow":326,"navLabel":327},"ch04","Bar graphs: drawing and reading","Chapter 04","4 Bar graphs",{"id":329,"type":43,"markdown":330},"bar-parts","A bar graph is the workhorse of data display. Get these parts right and anyone can read yours at a glance:\n\n- **Title** saying what the data is, where and when.\n- **Category axis** (usually horizontal) with a label on each bar.\n- **Value axis** (usually vertical) with a **uniform scale** that starts at **0**, and a label saying the units.\n- **Bars of equal width** with **equal gaps**; only the length of a bar carries information.\n\nBar graphs can stand up (vertical) or lie down (horizontal). Lying down is handy when category names are long, like the names of states.",{"id":332,"type":219,"title":333,"items":334},"steps-bar","Drawing a bar graph of enrolment in five classes (42, 38, 45, 36, 40 children)",[335,339,343,347,351],{"title":336,"tag":337,"text":338},"Find the largest value","45","The value axis must reach at least 45.",{"title":340,"tag":341,"text":342},"Choose a scale","1 unit = 5 children","On 10 cm of paper, 1 cm = 5 children makes 45 fit in 9 cm, leaving room. 1 cm = 1 child would need 45 cm.",{"title":344,"tag":345,"text":346},"Draw and label the axes","from 0","Mark 0, 5, 10… 50 evenly up the side. Label it \"Number of children\".",{"title":348,"tag":349,"text":350},"Draw the bars","equal width, equal gaps","Class 1: 42 ÷ 5 = 8.4 cm tall. Class 2: 7.6 cm. Class 3: 9 cm. Class 4: 7.2 cm. Class 5: 8 cm.",{"title":352,"tag":353,"text":354},"Add the title","what, where, when","\"Children enrolled in Classes 1–5, Government Primary School, 2026\".",{"id":356,"type":62,"caption":357,"columns":358,"rows":362},"table-enrol","Children enrolled in Classes 1–5 as a sideways bar graph. Scale: █ = 2 children (▌ = 1)",[359,360,361],"Class","Bar","Children",[363,367,371,374,378],[364,365,366],"Class 1","█████████████████████","42",[368,369,370],"Class 2","███████████████████","38",[372,373,337],"Class 3","██████████████████████▌",[375,376,377],"Class 4","██████████████████","36",[379,380,381],"Class 5","████████████████████","40",{"id":383,"type":244,"title":384,"problem":385,"steps":386},"we-bar-read","Reading the enrolment bar graph","Using the enrolment figures, answer: (a) Which class has the most children? (b) How many children are there in Classes 1–5 altogether? (c) By how much does the largest class exceed the smallest?",[387,388,389],"(a) The tallest bar is Class 3 with **45** children.","(b) 42 + 38 + 45 + 36 + 40 = **201 children**.","(c) Largest 45 (Class 3) − smallest 36 (Class 4) = **9 children**. This difference is the range of the data.",{"id":391,"type":47,"variant":392,"title":393,"markdown":394},"bar-zero","careful","Start the value axis at zero","If the value axis started at 30 instead of 0, Class 3's bar (45) would be 15 units tall and Class 4's (36) only 6 units: it would look **more than twice as big**, when really it is only a quarter bigger (45 ÷ 36 = 1.25). Bars compare lengths, so the lengths must start from zero. In Extend you will see how advertisers and news graphics sometimes break this rule, on purpose.",{"id":396,"type":254,"itemId":397,"prompt":398,"check":399,"hints":415,"feedback":416},"pr-scale","data-handling.understand-bar-scale","The largest value in your data is 2,400 and your graph has room for 12 grid lines. Which scale fits best?",{"kind":400,"options":401,"correct":414},"choice",[402,405,408,411],{"id":403,"label":404},"a","1 line = 10",{"id":406,"label":407},"b","1 line = 100",{"id":409,"label":410},"c","1 line = 200",{"id":412,"label":413},"d","1 line = 1,000",[409],[],{"correct":417,"incorrect":418},"Yes: 2,400 ÷ 200 = 12 lines exactly, so the tallest bar uses the whole height.","Divide the largest value by the scale and see if it fits in 12 lines. 2,400 ÷ 100 = 24 is too many; 2,400 ÷ 1,000 = 2.4 wastes the space.",{"id":420,"type":53,"title":421,"eyebrow":422,"navLabel":423},"ch05","The mean, exactly","Chapter 05","5 Mean",{"id":425,"type":43,"markdown":426},"mean-def","The **mean** (also called the **arithmetic mean**) of a set of numbers is\n\n**mean = sum of all the values ÷ number of values.**\n\nWhy does this give the fair share? Because adding collects everything into one heap, and dividing by the count shares the heap equally. The mean does not change the total: (mean) × (number of values) = (sum). That little fact is surprisingly powerful, and you will use it again and again.\n\nThe mean uses **every** value, which is its strength (nothing is ignored) and, as you will see, its weakness (one extreme value can drag it a long way).",{"id":428,"type":429,"items":430},"formulas-mean","formulas",[431,433,436],{"expression":39,"caption":432},"The fair share: add everything, then share equally.",{"expression":434,"caption":435},"sum = mean × count","Turn it round to find the total from the mean.",{"expression":437,"caption":438},"count = sum ÷ mean","Or find how many values there were.",{"id":440,"type":62,"caption":441,"columns":442,"rows":449},"table-delhi","New Delhi (Safdarjung): mean daily maximum temperature by month (IMD normals 1991–2020, rounded, °C)",[443,444,445,446,447,448],"Month","Jan","Feb","Mar","Apr","May",[450],[451,452,453,212,454,381],"°C","20","24","37",{"id":456,"type":244,"title":457,"problem":458,"steps":459},"we-mean-delhi","Mean monthly maximum temperature in Delhi","The mean daily maximum temperatures in New Delhi for the 12 months (IMD normals for 1991–2020, rounded to whole degrees) are 20, 24, 30, 37, 40, 39, 36, 34, 34, 33, 28, 23 °C. Find the mean.",[460,461,462,463,464],"Add all twelve values. Group them to make it easier: (20 + 24 + 30 + 37) + (40 + 39 + 36 + 34) + (34 + 33 + 28 + 23) = 111 + 149 + 118 = **378**.","Count: 12 months.","Mean = 378 ÷ 12 = **31.5 °C**.","Sense check: the mean must lie between the smallest (20) and largest (40) values, and 31.5 does. It is below the hot-season values and above the winter ones, as a fair share should be.","Notice that no month actually has 31.5 °C. The mean is a summary, not a member of the list.",{"id":466,"type":47,"variant":467,"title":468,"markdown":469},"mean-bounds","aha","The mean always sits between the extremes","Because it is a fair share, the mean can never be smaller than the smallest value or bigger than the largest. If you calculate a mean of 45 °C for Delhi's months, where the largest value is 40, you have made a slip. This quick check catches many errors.",{"id":471,"type":254,"itemId":472,"prompt":473,"check":474,"hints":477,"feedback":479},"pr-mean","data-handling.understand-mean-runs","A batter scored 45, 12, 0, 78 and 30 runs in five innings. What is the mean score per innings?",{"kind":258,"answer":475,"tolerance":170,"unit":476},33,"runs",[478],"The 0 still counts as an innings: divide by 5, not 4.",{"correct":480,"incorrect":481},"Correct: 45 + 12 + 0 + 78 + 30 = 165, and 165 ÷ 5 = 33.","Add all five scores, including the 0, then divide by 5.",{"id":483,"type":47,"variant":86,"title":484,"markdown":485},"zero-counts","\"Zeros don't count\"","A 0 is a real value. If a batter is out for 0, that innings happened and it counts in the count. Leaving it out would divide 165 by 4 and give 41.25, making the batter look better than they were. (Cricket's *batting average* has its own special rule about not-out innings; you will meet it in Deepen.)",{"id":487,"type":53,"title":488,"eyebrow":489,"navLabel":490},"ch06","The median: the middle value","Chapter 06","6 Median",{"id":492,"type":43,"markdown":493},"median-def","The **median** is the value in the **middle** of the data once it is **arranged in order** (ascending or descending, either works). Half the values are at or below it, half at or above it.\n\n- If the number of values **n is odd**, there is exactly one middle value: the **((n + 1) ÷ 2)th** value. For n = 7 that is the 4th.\n- If **n is even**, there are two middle values: the **(n ÷ 2)th** and the **(n ÷ 2 + 1)th**. The median is the **mean of these two**: add them and halve. For n = 12 that is the mean of the 6th and 7th values.\n\nSo the median, like the mean, might not be one of the data values when n is even.",{"id":495,"type":244,"title":496,"problem":497,"steps":498},"we-median-delhi","Median of Delhi's monthly maximum temperatures","Find the median of the 12 Delhi values: 20, 24, 30, 37, 40, 39, 36, 34, 34, 33, 28, 23 °C.",[499,500,501,502,503],"Order them: 20, 23, 24, 28, 30, 33, 34, 34, 36, 37, 39, 40.","n = 12 is even, so the middle two are the 6th and 7th values.","6th = 33, 7th = 34.","Median = (33 + 34) ÷ 2 = 67 ÷ 2 = **33.5 °C**.","Compare with the mean, 31.5 °C. The median is higher, because the cold winter months (20, 23, 24) drag the mean down, while the median only cares about the middle position.",{"id":505,"type":244,"title":506,"problem":507,"steps":508},"we-median-odd","Median with an odd count, and a repeated value","The weekly pocket money of 9 children (₹) is 50, 100, 20, 50, 75, 200, 50, 60, 40. Find the median.",[509,510,511,512],"Order: 20, 40, 50, 50, 50, 60, 75, 100, 200.","n = 9 is odd, so the median is the ((9 + 1) ÷ 2) = 5th value.","Count along: 20 (1st), 40 (2nd), 50 (3rd), 50 (4th), **50 (5th)**.","Median = **₹50**. Repeated values are all kept in the ordered list; do not drop duplicates.",{"id":514,"type":47,"variant":86,"title":515,"markdown":516},"median-dup","\"Cross out repeated numbers before finding the middle\"","Every child's value counts, even if two children have the same amount. If you removed duplicates from the pocket-money list you would get 20, 40, 50, 60, 75, 100, 200, whose middle is 60: wrong. The median describes the **people** (or things), so each one must stay in the line.",{"id":518,"type":254,"itemId":519,"prompt":520,"check":521,"hints":522,"feedback":525},"pr-median-even","data-handling.understand-median-even","Find the median of these 8 test marks: 14, 18, 11, 20, 15, 17, 12, 19.",{"kind":258,"answer":259,"tolerance":170},[523,524],"n = 8 is even.","Add the two middle values and halve.",{"correct":526,"incorrect":527},"Right: ordered 11, 12, 14, 15, 17, 18, 19, 20. The 4th and 5th are 15 and 17, and (15 + 17) ÷ 2 = 16.","Order them first. With 8 values the median is halfway between the 4th and 5th.",{"id":529,"type":53,"title":530,"eyebrow":531,"navLabel":532},"ch07","The mode: the most common value","Chapter 07","7 Mode",{"id":534,"type":43,"markdown":535},"mode-def","The **mode** is the value with the **highest frequency**. It is the only average you can find for categorical data (the most popular snack, the most common blood group), and it is also useful for numerical data where only real, existing values make sense (shoe sizes, clothing sizes, number of people in an auto).\n\nThree special cases:\n\n- **Two modes (bimodal):** if two values tie for the highest frequency, both are modes. In the marks 3, 5, 5, 7, 8, 8, 9, both 5 and 8 appear twice and everything else once: modes 5 and 8.\n- **More than two:** three or more tied values are all modes; the data is **multimodal**, and the mode is not very informative. Chennai's twelve monthly maximum temperatures, rounded to whole degrees (30, 31, 33, 35, 37, 37, 36, 35, 34, 33, 30, 29 °C), have four values tied at two appearances each — 30, 33, 35 and 37 — so the mode tells you almost nothing.\n- **No mode:** if every value appears exactly once, no value is more common than any other, and we say there is **no mode**. (Some books say every value is a mode; this lesson follows the more common school convention of \"no mode\".)",{"id":537,"type":62,"caption":538,"columns":539,"rows":543},"table-modes","Mode in five situations",[540,541,542],"Data","Frequencies","Mode",[544,548,552,556,560],[545,546,547],"Shoe sizes 4, 5, 5, 6, 5, 7","5 appears 3 times","5",[549,550,551],"Marks 3, 5, 5, 7, 8, 8, 9","5 and 8 each appear twice","5 and 8 (bimodal)",[553,554,555],"Chennai monthly maxima (12 values)","30, 33, 35 and 37 each appear twice","Four modes (multimodal)",[557,558,559],"Marks 12, 15, 18, 20","each appears once","No mode",[561,562,563],"Blood groups O, A, O, B, AB, O, A","O appears 3 times","O (categorical)",{"id":565,"type":244,"title":566,"problem":567,"steps":568},"we-mode-table","Mode from a frequency table","In the siblings table, the frequencies were 0 → 4, 1 → 12, 2 → 9, 3 → 4, 4 → 1. What is the mode?",[569,570,571],"Look for the **largest frequency**, not the largest value: it is 12.","The value with frequency 12 is **1 sibling**. So the mode is 1.","A very common slip is to answer \"12\". But 12 is *how many children* chose that value, not the value itself.",{"id":573,"type":47,"variant":86,"title":574,"markdown":575},"mode-slip","\"The mode is the biggest frequency\"","The mode is a **value** from the data (1 sibling, size 5, samosa). The frequency is **how often** it occurs. Always answer with the value, and use the units of the data, as in \"the modal shoe size is 5\", not \"the mode is 3 children\".",{"id":577,"type":254,"itemId":578,"prompt":579,"check":580,"hints":590,"feedback":591},"pr-mode","data-handling.understand-mode-bi","What is the mode of 3, 7, 2, 7, 9, 3, 5?",{"kind":400,"options":581,"correct":589},[582,584,585,587],{"id":403,"label":583},"7",{"id":406,"label":208},{"id":409,"label":586},"3 and 7",{"id":412,"label":588},"There is no mode",[409],[],{"correct":592,"incorrect":593},"Yes: 3 and 7 each appear twice, more than any other value, so the data is bimodal.","Count each value: 3 twice, 7 twice, the others once. A tie at the top gives two modes.",{"id":595,"type":53,"title":596,"eyebrow":597,"navLabel":598},"ch08","Range: how spread out?","Chapter 08","8 Range",{"id":600,"type":43,"markdown":601},"range-def","The **range** is the difference between the **largest** and **smallest** values:\n\n**range = maximum − minimum.**\n\nThe range is not an average. Averages tell you where the data is *centred*; the range tells you how *spread out* it is. Two data sets can have similar averages but very different spreads, and that difference can matter enormously.\n\nCompare the two cities. Delhi's monthly maxima run from 20 °C (January) to 40 °C (May): range **20 °C**. Chennai's run from 29 °C (December) to 37 °C (May and June): range only **8 °C**. Their means are fairly close (Delhi 31.5 °C, Chennai ≈ 33.3 °C), and their medians are exactly equal (33.5 °C each)! Yet anyone who has lived in both knows the difference: Delhi has a real winter and a scorching summer; Chennai is warm to hot all year, because the sea next to it evens out the temperature.",{"id":603,"type":62,"caption":604,"columns":605,"rows":609},"table-cities","Delhi and Chennai compared (mean daily maximum by month, IMD normals 1991–2020, rounded, °C)",[606,607,608],"Measure","Delhi","Chennai",[610,614,618,621,624,628,632],[611,612,613],"Sum of 12 months","378","400",[615,616,617],"Mean","378 ÷ 12 = 31.5","400 ÷ 12 ≈ 33.3",[619,620,620],"Median","(33 + 34) ÷ 2 = 33.5",[542,622,623],"34","30, 33, 35 and 37",[625,626,627],"Maximum","40 (May)","37 (May and Jun)",[629,630,631],"Minimum","20 (Jan)","29 (Dec)",[633,634,635],"Range","40 − 20 = 20","37 − 29 = 8",{"id":637,"type":91,"component":638,"componentVersion":5,"config":639,"objective":662,"textAlternative":663,"help":664},"lab-cities","data-lab",{"datasets":640,"valueRange":660,"step":5},[641,654],{"label":642,"unit":451,"values":643},"New Delhi (Safdarjung): mean daily maximum by month",[644,645,646,647,648,649,650,651,651,475,652,653],20,24,30,37,40,39,36,34,28,23,{"label":655,"unit":451,"values":656},"Chennai (Nungambakkam): mean daily maximum by month",[646,657,475,658,647,647,650,658,651,475,646,659],31,35,29,{"min":661,"max":18},15,"Switch between Delhi and Chennai and compare their mean, median, mode and range.","Two datasets of 12 monthly values each, shown as dot plots from 15 to 45 °C.\n\n**Delhi:** 20, 24, 30, 37, 40, 39, 36, 34, 34, 33, 28, 23. Mean 31.5, median 33.5, mode 34, range 20. The dots are spread wide, from 20 to 40.\n\n**Chennai:** 30, 31, 33, 35, 37, 37, 36, 35, 34, 33, 30, 29. Mean ≈ 33.3, median 33.5, four modes (30, 33, 35 and 37), range 8. The dots are bunched between 29 and 37.\n\nThe centres are almost the same; the spreads are very different. Try dragging Delhi's January value up from 20 to 29: the mean rises by 9 ÷ 12 = 0.75 to 32.25, the range shrinks to 40 − 23 = 17, and the median does not move at all.",{"simplerExplanation":665},"Both cities are about equally hot on average, but Delhi swings between cold and very hot, while Chennai stays warm all year. The range shows the swing.",{"id":667,"type":47,"variant":668,"title":669,"markdown":670},"range-limit","nuance","The range only looks at two values","The range is quick, but it uses only the maximum and the minimum and ignores everything in between. One unusual value, like a single freak hailstorm day, can make the range huge even if all the other days were similar. In Deepen you will see a sturdier measure of spread that ignores the extremes.",{"id":672,"type":254,"itemId":673,"prompt":674,"check":675,"hints":678,"feedback":679},"pr-range","data-handling.understand-range-bill","A family's monthly electricity use over a year (units) was lowest at 165 in February and highest at 380 in June. What is the range?",{"kind":258,"answer":676,"tolerance":170,"unit":677},215,"units",[],{"correct":680,"incorrect":681},"Right: 380 − 165 = 215 units. Summer air-coolers and fans make a big difference!","Range = maximum − minimum = 380 − 165.",{"id":683,"type":53,"title":684,"eyebrow":685,"navLabel":686},"ch09","Averages straight from a frequency table","Chapter 09","9 From a table",{"id":688,"type":43,"markdown":689},"table-avg","Suppose 25 children took a quiz marked out of 10. Rather than list 25 numbers, the teacher gives a frequency table. You can find every summary **without** writing out the list.\n\n- **Mean:** each row contributes *value × frequency* to the sum. Add a column **f × x**; the mean is (sum of f × x) ÷ (sum of f).\n- **Median:** use a running total (cumulative frequency) to find which row contains the middle position.\n- **Mode:** the row with the largest frequency.\n- **Range:** the largest value with a non-zero frequency minus the smallest.",{"id":691,"type":62,"caption":692,"columns":693,"rows":697},"table-marks","Quiz marks (out of 10) of 25 children",[694,695,696,191],"Mark x","Frequency f","f × x",[698,700,702,705,708,711,714,716],[196,198,699,198],"1 × 4 = 4",[547,203,701,208],"2 × 5 = 10",[703,196,704,583],"6","4 × 6 = 24",[583,703,706,707],"6 × 7 = 42","13",[709,583,710,452],"8","7 × 8 = 56",[205,208,712,713],"3 × 9 = 27","23",[33,203,715,206],"2 × 10 = 20",[214,717,718,215],"**25**","**183**",{"id":720,"type":244,"title":721,"problem":722,"steps":723},"we-table-avg","All four summaries from the quiz table","Use the quiz-marks table to find the mean, median, mode and range.",[724,725,726,727,728],"**Mean:** sum of f × x = 4 + 10 + 24 + 42 + 56 + 27 + 20 = 183. Number of children = 25. Mean = 183 ÷ 25 = **7.32 marks**.","**Median:** n = 25 is odd, so the median is the 13th value. The running totals are 1, 3, 7, 13, … The 13th child is the last one in the row for mark 7. Median = **7**.","**Mode:** the largest frequency is 7, in the row for mark 8. Mode = **8**.","**Range:** highest mark 10 − lowest mark 4 = **6**.","All three averages differ: mean 7.32, median 7, mode 8. That is normal. Each answers a different question.",{"id":730,"type":91,"component":638,"componentVersion":5,"config":731,"objective":759,"textAlternative":760},"lab-marks",{"datasets":732,"valueRange":744,"step":5,"challenges":745},[733],{"label":734,"unit":735,"values":736},"Quiz marks of 25 children (out of 10)","marks",[737,738,738,739,739,739,739,740,740,740,740,740,740,741,741,741,741,741,741,741,742,742,742,743,743],4,5,6,7,8,9,10,{"min":170,"max":743},[746,750,753,756],{"measure":747,"target":748,"prompt":749},"mean",7.4,"Two children asked for a recheck. Raise marks so the class mean becomes 7.4.",{"measure":751,"target":741,"prompt":752},"median","Change as few marks as you can so the median becomes 8.",{"measure":754,"target":740,"prompt":755},"mode","Make 7 the only mode.",{"measure":757,"target":738,"prompt":758},"range","Make the range 5 by changing one child's mark.","Edit quiz marks and see how the mean, median, mode and range change, then hit the targets.","A dot plot of 25 quiz marks: one 4, two 5s, four 6s, six 7s, seven 8s, three 9s and two 10s. Sum 183, mean 183 ÷ 25 = **7.32**, median **7** (13th value), mode **8**, range **6**.\n\nChallenges:\n\n1. **Mean 7.4:** the sum must become 7.4 × 25 = 185, so the marks must rise by 2 in total (e.g. two children go up by 1 each).\n2. **Median 8:** the 13th ordered value must be 8. At present 13 children have 7 or less; move at least one of them up to 8 or more, e.g. one 7 → 8 (then 12 children are at 7 or below and the 13th is an 8).\n3. **Mode 7 only:** 7 needs more children than 8; move one 8 to 7 (then 7 has 7 children, 8 has 6).\n4. **Range 5:** change the single 4 to a 5 (10 − 5 = 5).",{"id":762,"type":254,"itemId":763,"prompt":764,"check":765,"hints":768,"feedback":770},"pr-table-mean","data-handling.understand-table-mean","A survey of 20 homes counted mobile phones: 1 phone in 3 homes, 2 phones in 8 homes, 3 phones in 6 homes, 4 phones in 3 homes. What is the mean number of phones per home?",{"kind":258,"answer":766,"tolerance":767},2.45,0.01,[769],"Divide by the total frequency, 20.",{"correct":771,"incorrect":772},"Correct: f × x gives 3 + 16 + 18 + 12 = 49 phones in 20 homes, and 49 ÷ 20 = 2.45.","Multiply each number of phones by how many homes have it, add those up (49), then divide by the 20 homes, not by 4 rows.",{"id":774,"type":91,"component":775,"componentVersion":5,"config":776,"objective":810,"textAlternative":811},"lab-sprint","arith-sprint",{"operations":777,"ranges":779,"rounds":743,"secondsTotal":170,"estimateFirst":784,"wordProblems":785},[778],"÷",{"a":780,"b":783},{"min":781,"max":782},2,12,{"min":781,"max":743},false,[786,789,792,795,797,801,804,807],{"prompt":787,"answer":661,"operation":778,"unit":788},"Five friends have 12, 15, 9, 18 and 21 marbles. What is the mean number of marbles?","marbles",{"prompt":790,"answer":648,"operation":778,"unit":791},"A shop sold 240 samosas over 6 days. What is the mean number sold per day?","samosas",{"prompt":793,"answer":657,"operation":778,"unit":794},"Four runs of a school bus took 32, 28, 35 and 29 minutes. What was the mean time?","minutes",{"prompt":796,"answer":649,"operation":778,"unit":476},"A batter made 312 runs in 8 innings, all out. What is the mean score per innings?",{"prompt":798,"answer":799,"operation":800},"The mean of 6 numbers is 14. What is their sum?",84,"×",{"prompt":802,"answer":741,"operation":778,"unit":803},"Rainfall on 7 days was 0, 12, 5, 30, 0, 8 and 1 mm. What was the mean daily rainfall?","mm",{"prompt":805,"answer":740,"operation":778,"unit":806},"Three bags weigh 4 kg, 7 kg and 10 kg. What is the mean mass?","kg",{"prompt":808,"answer":809,"operation":778,"unit":677},"A family used 1,080 units of electricity in 6 months. What was the mean use per month?",180,"Practise mean calculations as quick word problems and warm-up divisions.","An untimed sprint of 10 division warm-ups, followed by mean word problems. Answers:\n\n- Marbles 12, 15, 9, 18, 21: sum 75, mean 75 ÷ 5 = **15**.\n- 240 samosas over 6 days: **40** a day.\n- Bus times 32, 28, 35, 29: sum 124, mean **31** minutes.\n- 312 runs in 8 innings: **39**.\n- Mean 14 of 6 numbers: sum = 14 × 6 = **84**.\n- Rainfall 0, 12, 5, 30, 0, 8, 1: sum 56, mean 56 ÷ 7 = **8 mm** (zeros count!).\n- Bags 4, 7, 10 kg: mean **7 kg**.\n- 1,080 units in 6 months: **180 units** a month.",{"id":813,"type":53,"title":814,"eyebrow":815,"navLabel":816},"ch10","Mix-ups, checks and a round-up","Chapter 10","10 Mix-ups",{"id":818,"type":62,"caption":819,"columns":820,"rows":824},"table-mixups","Common mix-ups and how to avoid them",[821,822,823],"Mix-up","Why it is wrong","Fix",[825,829,833,837,841,845,849,853],[826,827,828],"Finding the median without ordering","The middle of an unsorted list is just whoever was written in the middle.","Always order first.",[830,831,832],"Dropping repeated values","Each repeat is a separate person or thing.","Keep every value in the ordered list.",[834,835,836],"Giving the frequency as the mode","The mode is a value; the frequency is how often it occurs.","Answer with the value and its units.",[838,839,840],"Dividing by the number of rows in a frequency table","Rows are values, not observations.","Divide by the total frequency.",[842,843,844],"Leaving out zeros","A zero is a real observation.","Count zeros in the count.",[846,847,848],"Range = largest value","The range is a difference.","Subtract the smallest value.",[850,851,852],"Mean of categories","You cannot add colours or languages.","Use the mode for categorical data.",[854,855,856],"A mean outside the data","A fair share must lie between min and max.","Recheck the sum or the count.",{"id":858,"type":91,"component":859,"componentVersion":5,"config":860,"objective":883,"textAlternative":884},"lab-match-mean","match-pairs",{"prompt":861,"mode":862,"pairs":863},"Match each data set to its mean.","connect",[864,866,868,870,873,875,878,880],{"a":865,"b":547},"2, 4, 6, 8",{"a":867,"b":452},"10, 20, 30",{"a":869,"b":208},"1, 1, 1, 9",{"a":871,"b":872},"5, 5, 5, 5, 5","5 as well: all equal",{"a":874,"b":196},"0, 0, 12",{"a":876,"b":877},"7, 8","7.5",{"a":879,"b":205},"3, 6, 9, 12, 15",{"a":881,"b":882},"100, 0","50","Match each small data set to its mean by finding sum ÷ count.","Eight data sets to match with their means (sum ÷ count):\n\n- 2, 4, 6, 8 → 20 ÷ 4 = **5**\n- 10, 20, 30 → 60 ÷ 3 = **20**\n- 1, 1, 1, 9 → 12 ÷ 4 = **3**\n- 5, 5, 5, 5, 5 → **5** (all values equal: the mean is that value)\n- 0, 0, 12 → 12 ÷ 3 = **4**\n- 7, 8 → 15 ÷ 2 = **7.5**\n- 3, 6, 9, 12, 15 → 45 ÷ 5 = **9**\n- 100, 0 → **50**\n\nTwo sets share a mean of 5: a spread-out set (2, 4, 6, 8) and a set with no spread at all. The mean alone does not tell you the spread.",{"id":886,"type":887,"prompt":888,"options":889,"explanation":898},"predict-order","prediction","Here are 6 numbers: 8, 3, 8, 5, 10, 2. Without calculating fully, which is the **largest**: the mean, the median or the mode?",[890,892,894,896],{"id":403,"label":891},"The mean",{"id":406,"label":893},"The median",{"id":409,"label":895},"The mode",{"id":412,"label":897},"All three are equal","**The mode (8) is the largest.** The mean is 36 ÷ 6 = 6. Ordered, the data is 2, 3, 5, 8, 8, 10, so the median is (5 + 8) ÷ 2 = 6.5. The mode is 8, the only repeated value. Different averages can come out quite different, especially in small data sets.",{"id":900,"type":901,"title":902,"terms":903},"glossary-understand","glossary","Words from this layer",[904,908,912,916,920,923,927,930,933,937,940,944,947],{"term":905,"meaning":906,"example":907},"categorical data","Data that sorts things into named groups (categories), such as colour or language. It can be counted but not added.","Blood groups A, B, AB, O.",{"term":909,"meaning":910,"example":911},"numerical data","Data made of numbers that count or measure something, so adding and averaging make sense.","Heights in cm.",{"term":913,"meaning":914,"example":915},"discrete data","Numerical data that can only take separate values, usually whole numbers from counting.","Number of siblings.",{"term":917,"meaning":918,"example":919},"continuous data","Numerical data from measuring, which can take any value in a range.","Time to run 100 m: 15.73 s.",{"term":921,"meaning":922},"primary data","Data you collect yourself for your own question.",{"term":924,"meaning":925,"example":926},"secondary data","Data collected by someone else that you reuse.","IMD rainfall records.",{"term":928,"meaning":929},"axis","One of the two reference lines of a graph: one for categories or values along the bottom, one for numbers up the side.",{"term":931,"meaning":932},"arithmetic mean","The sum of the values divided by how many there are; the full name of the mean.",{"term":934,"meaning":935,"example":936},"bimodal","Having two modes: two values tie for the highest frequency.","3, 3, 5, 7, 7 → modes 3 and 7.",{"term":938,"meaning":939},"no mode","The situation when every value appears the same number of times (for example, each once), so none is most common.",{"term":941,"meaning":942,"example":943},"cumulative frequency","A running total of frequencies, adding each row to those before it.","4, 16, 25, 29, 30.",{"term":945,"meaning":946},"spread","How far apart the values of a data set are; the range is one measure of spread.",{"term":948,"meaning":949,"example":950},"ascending order","Arranged from smallest to largest.","2, 5, 7, 9.",{"id":952,"type":953,"title":954,"questions":955},"quiz-understand","quiz","Methods check",[956,969,981,991,1000,1012,1022,1034,1047,1059],{"itemId":957,"prompt":958,"options":959,"correct":409,"why":968},"data-handling.understand-q-kind","Which of these is continuous numerical data?",[960,962,964,966],{"id":403,"label":961},"Number of goals scored",{"id":406,"label":963},"Favourite fruit",{"id":409,"label":965},"Mass of a watermelon",{"id":412,"label":967},"Roll number","Mass is measured and can take any value, e.g. 4.37 kg. Goals are counted (discrete); fruit and roll numbers are categories.",{"itemId":970,"prompt":971,"options":972,"correct":406,"why":980},"data-handling.understand-q-mean","Find the mean of 12, 15, 18, 21, 24.",[973,975,977,979],{"id":403,"label":974},"15",{"id":406,"label":976},"18",{"id":409,"label":978},"19",{"id":412,"label":290},"Sum 90, count 5, mean 90 ÷ 5 = 18.",{"itemId":982,"prompt":983,"options":984,"correct":409,"why":990},"data-handling.understand-q-median-even","What is the median of 4, 9, 1, 7, 3, 10?",[985,986,987,989],{"id":403,"label":583},{"id":406,"label":547},{"id":409,"label":988},"5.5",{"id":412,"label":709},"Ordered: 1, 3, 4, 7, 9, 10. The middle two are 4 and 7; (4 + 7) ÷ 2 = 5.5.",{"itemId":992,"prompt":993,"options":994,"correct":409,"why":999},"data-handling.understand-q-mode","What is the mode of 2, 4, 6, 8, 10?",[995,996,997,998],{"id":403,"label":703},{"id":406,"label":33},{"id":409,"label":559},{"id":412,"label":203},"Every value appears once, so no value is most common.",{"itemId":1001,"prompt":1002,"options":1003,"correct":406,"why":1011},"data-handling.understand-q-freq-mean","In a table, 2 children scored 5, 3 scored 6 and 5 scored 8. What is the mean score?",[1004,1006,1008,1010],{"id":403,"label":1005},"6.4",{"id":406,"label":1007},"6.8",{"id":409,"label":1009},"6.33",{"id":412,"label":978},"f × x: 10 + 18 + 40 = 68, total frequency 10, mean 68 ÷ 10 = 6.8.",{"itemId":1013,"prompt":1014,"options":1015,"correct":409,"why":1021},"data-handling.understand-q-range","The range of a data set is 12 and its smallest value is 25. What is its largest value?",[1016,1017,1018,1019],{"id":403,"label":707},{"id":406,"label":200},{"id":409,"label":454},{"id":412,"label":1020},"300","Range = max − min, so max = 25 + 12 = 37.",{"itemId":1023,"prompt":1024,"options":1025,"correct":406,"why":1033},"data-handling.understand-q-mode-value","A table shows shoe size 6 with frequency 11, the highest frequency. The mode is…",[1026,1028,1029,1031],{"id":403,"label":1027},"11",{"id":406,"label":703},{"id":409,"label":1030},"17",{"id":412,"label":1032},"Not enough information","The mode is the value (size 6), not its frequency (11).",{"itemId":1035,"prompt":1036,"options":1037,"correct":403,"why":1046},"data-handling.understand-q-zero","Rainfall on 5 days: 0, 0, 10, 20, 0 mm. The mean daily rainfall is…",[1038,1040,1042,1044],{"id":403,"label":1039},"6 mm",{"id":406,"label":1041},"15 mm",{"id":409,"label":1043},"10 mm",{"id":412,"label":1045},"0 mm","Sum 30 over 5 days: 6 mm. Dry days count.",{"itemId":1048,"prompt":1049,"options":1050,"correct":409,"why":1058},"data-handling.understand-q-picto","A pictograph key is ☂ = 8 rainy days. How many days does 3½ ☂ show?",[1051,1052,1054,1056],{"id":403,"label":453},{"id":406,"label":1053},"26",{"id":409,"label":1055},"28",{"id":412,"label":1057},"35","3 × 8 = 24, plus half of 8 = 4: 28 days.",{"itemId":1060,"prompt":1061,"options":1062,"correct":406,"why":1071},"data-handling.understand-q-axis","Why should the value axis of a bar graph start at 0?",[1063,1065,1067,1069],{"id":403,"label":1064},"It looks neater",{"id":406,"label":1066},"So bar lengths are in the same ratio as the values",{"id":409,"label":1068},"Because negative numbers are not allowed",{"id":412,"label":1070},"It does not matter","If the axis starts higher, short bars shrink more than long ones and differences look exaggerated.",{"id":1073,"type":1074,"title":1075,"points":1076},"cheat-understand","summary","Cheat sheet",[1077,1078,1079,1080,1081,1082,1083,1084,1085,1086],"**Categorical** data names groups (use the mode, bar graphs, pictographs). **Numerical** data counts (**discrete**) or measures (**continuous**).","A digit label (PIN code, jersey number) is still categorical: adding it makes no sense.","**Frequency table:** list values in order, tally once, check the total, add a running total.","**Pictograph key:** choose it so the biggest row has under about 10 symbols and values divide into whole or half symbols.","**Bar graph:** title, labelled axes, uniform scale starting at 0, equal widths and gaps.","**Mean** = sum ÷ count; sum = mean × count. It always lies between min and max. Zeros count.","**Median:** order first. n odd → ((n + 1) ÷ 2)th value. n even → mean of the (n ÷ 2)th and (n ÷ 2 + 1)th values.","**Mode:** value with the highest frequency; two modes if tied; no mode if all values appear once.","**Range** = max − min: a measure of spread, not an average.","**From a table:** mean = Σ(f × x) ÷ Σf; median from the running total; mode = row with the largest f.",{"id":1088,"type":1089,"prompt":1090},"reflect-understand","reflection","Delhi and Chennai have the same median monthly maximum temperature, 33.5 °C. Write two sentences for a friend explaining why \"the same average\" does not mean \"the same weather\", using the word range.",{"id":1092,"type":1093,"conceptId":1094,"relation":1095,"explanation":1096},"conn-ops","connection","order-of-operations","helps_understand","Mean = (sum) ÷ (count): the brackets matter. 2 + 4 + 6 ÷ 3 is not the mean of 2, 4 and 6.",{"id":1098,"type":1093,"conceptId":1099,"relation":1095,"explanation":1100},"conn-props","properties-of-numbers","Grouping values cleverly to add them uses the commutative and associative properties of addition.",{"id":1102,"type":1093,"conceptId":1103,"relation":1095,"explanation":1104},"conn-four","four-operations","Frequency tables use multiplication (f × x) as quick repeated addition.",{"id":1106,"type":1107,"sourceIds":1108},"sources-understand","sources",[1109,1110,1111,1112,1113,1114],"data-handling-ncert-class6-gp","data-handling-ncert-class7","data-handling-khan-summarizing","data-handling-mathsisfun-central","data-handling-imd","data-handling-imd-normals",[1109,1110,1111,1112,1113,1114],"needs_review",{"generatedBy":1118,"notes":1119},"claude-code","Draft generated with Python-checked statistics; pending owner review.","051ea35d6871a8ee18276929654e7ae374e3dbf73e98debf137a391cf57ff3e6",{"component:sort-game@1":1122,"logic:practice":1123,"component:data-lab@1":1124,"component:arith-sprint@1":1125,"component:match-pairs@1":1126,"source:data-handling-imd":1127,"source:data-handling-imd-normals":1128,"source:data-handling-khan-summarizing":1129,"source:data-handling-mathsisfun-central":1130,"source:data-handling-ncert-class6-gp":1131,"source:data-handling-ncert-class7":1132},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","466896cc37735f48db03875fe9c9ce42fc8bcb7e5f937c9779d70513703b91bd","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","834b555363edd0d5f3a42f5870d2acb5a33c3af5e40ecb43649732edd4cdce2c","f8d2cede5dff9df165f0ad49c28625d281417b1abac9405104cd7e3ff5c50f88","d3f02b5bb1750887d4c5e469441199469eba3f40ba38f08a0f15e6dc123789da","0e1e3ad9dc9f84bb51edeb70e657591671f3d8e7326c34f58c2de0c70e823635","3d89b7ad9db743739787967056153b3afc3772805c18a2f4b73fd681b870cc48","73c4938c33c83e313b3e69ea6d4b3fcd121cfeab65eda7e07e1df7901cfaea2c",{"state":1134,"reviewer":1135,"selfReview":1136,"reviewedAt":1137,"method":1138},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597840]