[{"data":1,"prerenderedAt":977},["ShallowReactive",2],{"layer:data-handling:investigate":3},{"layer":4,"contentHash":958,"dependencyHashes":959,"approval":970,"releaseId":976},{"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":953,"reviewStatus":954,"authoring":955},1,"data-handling","en","investigate","What happens if…? Experiments with averages","Predict, change the data, and test: outliers, shifts, missing values and datasets built to order","Treat averages like a science experiment. Predict what adding a value, an outlier, or a change to every value does to the mean, median, mode and range, then test it in the labs. Build data sets to order, hunt missing values and compare real Indian data.",[13,14,15,16,17],"Predict and test how adding or removing a value changes the mean, median, mode and range.","Explain why an outlier moves the mean a lot but the median very little.","Predict the effect of adding a constant to every value, or multiplying every value.","Build a data set that has a given mean, median, mode and range, and find missing values.","Decide whether statements about averages are always, sometimes or never true, with examples.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Mean, median, mode, range (Understand)",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","5 data labs and a what-changes sort",{"label":38,"value":39},"Style","Predict first, then test",[41,45,51,57,60,78,109,112,155,168,173,176,189,194,223,228,248,253,266,279,315,320,393,404,409,412,450,463,477,482,487,490,503,528,538,543,546,559,562,588,592,597,600,620,636,640,645,648,657,666,675,687,698,703,706,736,749,753,756,761,764,798,912,926,932,937,942],{"id":42,"type":43,"markdown":44},"intro","prose","Scientists do not just learn facts; they **poke** things to see what happens. In this layer you will poke data. What if a new value joins? What if one value is enormous? What if everyone gets 5 more marks? What if I double everything?\n\nFor each experiment: **predict first** (commit to an answer), then **test** in a lab or by calculating, then **explain** what you saw. Some results will surprise you, and those surprises are exactly where the understanding is hiding.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"method","callout","observation","The investigator's loop","1. **Predict:** what will happen to the mean? The median? The mode? The range?\n2. **Test:** change the data in a lab, or calculate on paper.\n3. **Explain:** why did it happen? Use the ideas of *fair share* (mean) and *middle position* (median).\n4. **Generalise:** is it always true, or only for this data? Try to find a counterexample.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","A new value joins the data","Chapter 01","1 Adding a value",{"id":58,"type":43,"markdown":59},"inn-setup","Priya, a school cricketer, scored 34, 12, 56, 8, 42, 26, 32 in her last 7 innings (she was out every time). Her mean is 210 ÷ 7 = **30** runs, and her median is **32** (ordered: 8, 12, 26, 32, 34, 42, 56).\n\nShe is about to bat again. Let us experiment with what her next score does to her averages.",{"id":61,"type":62,"prompt":63,"options":64,"explanation":77},"predict-add-30","prediction","In her 8th innings Priya scores exactly **30**, the same as her mean. What happens to her mean and median?",[65,68,71,74],{"id":66,"label":67},"a","Both stay the same",{"id":69,"label":70},"b","Mean stays 30; median changes",{"id":72,"label":73},"c","Mean changes; median stays 32",{"id":75,"label":76},"d","Both change","**The mean stays 30, but the median changes to 31.** The new total is 210 + 30 = 240 over 8 innings: 240 ÷ 8 = 30. A value equal to the mean adds exactly one more fair share, so the fair share cannot change. But the median is about *position*: now there are 8 values (8, 12, 26, **30, 32**, 34, 42, 56), so the median is the mean of the 4th and 5th, (30 + 32) ÷ 2 = **31**.",{"id":79,"type":80,"caption":81,"columns":82,"rows":88},"table-add","table","What Priya's 8th score does to her averages (starting mean 30, median 32)",[83,84,85,86,87],"8th score","New total","New mean","New median","Range",[89,95,100,104],[90,91,92,93,94],"0","210","26.25","29","56",[96,97,96,98,99],"30","240","31","48",[101,102,98,103,99],"38","248","33",[105,106,107,103,108],"100","310","38.75","92",{"id":110,"type":43,"markdown":111},"add-rule","The pattern in the table is a rule you can rely on:\n\n- A new value **above** the mean pulls the mean **up**.\n- A new value **below** the mean pulls it **down**.\n- A new value **equal to** the mean leaves it **unchanged**.\n\nAnd *how far* it pulls depends on how far the new value is from the mean, shared over the new count. A score of 38 is 8 above the mean of 30, so the mean rises by 8 ÷ 8 = 1, to 31. A score of 100 is 70 above, so the mean rises by 70 ÷ 8 = 8.75, to 38.75.\n\nThe median moves at most a little: it only shifts by half a step along the ordered list, however big or small the new value is.",{"id":113,"type":114,"component":115,"componentVersion":5,"config":116,"objective":149,"textAlternative":150,"help":151},"lab-innings","interactive","data-lab",{"datasets":117,"valueRange":130,"step":133,"challenges":134},[118],{"label":119,"unit":120,"values":121},"Priya's scores in 8 innings","runs",[122,123,124,125,126,127,128,129],34,12,56,8,42,26,32,30,{"min":131,"max":132},0,80,2,[135,139,142,145],{"measure":136,"target":137,"prompt":138},"mean",33,"Change her 8th score (30) so that her mean becomes 33.",{"measure":136,"target":140,"prompt":141},27,"What 8th score would drag her mean down to 27?",{"measure":143,"target":137,"prompt":144},"median","Make the median 33 by changing only the 8th score.",{"measure":146,"target":147,"prompt":148},"range",70,"Make the range 70 runs.","Change Priya's 8th innings and watch how far each score pulls her mean and median.","A dot plot of 8 innings: 34, 12, 56, 8, 42, 26, 32 and a changeable 8th score, starting at 30. Mean 30, median 31, range 48.\n\nChallenges:\n\n1. **Mean 33:** the total must be 33 × 8 = 264, so the 8th score must be 264 − 210 = **54**.\n2. **Mean 27:** total 27 × 8 = 216, so the 8th score must be 216 − 210 = **6**. Try asking for a mean of 26: the 8th score would have to be −2, which is impossible, so you would need to change an earlier score too.\n3. **Median 33:** with the 8th score at 34 or more, the middle two are 32 and 34, so the median is 33.\n4. **Range 70:** the smallest score is 8, so make a score of 78.\n\nEach extra run in one innings lifts the mean by 1 ÷ 8 = 0.125.",{"hints":152},[153,154],"New mean × 8 = new total. Subtract the other seven scores (210).","The median depends on which two values are in the middle of the ordered list.",{"id":156,"type":157,"itemId":158,"prompt":159,"check":160,"hints":163,"feedback":165},"pr-next","practice","data-handling.investigate-next-score","A bowler's mean over 4 matches is 3 wickets. How many wickets must she take in her 5th match to raise her mean to 3.4?",{"kind":161,"answer":162,"tolerance":131},"number",5,[164],"sum = mean × count.",{"correct":166,"incorrect":167},"Right: she needs a total of 3.4 × 5 = 17 wickets, and she already has 3 × 4 = 12, so 17 − 12 = 5.","Work with totals: old total = 3 × 4 = 12. New total must be 3.4 × 5 = 17.",{"id":169,"type":53,"title":170,"eyebrow":171,"navLabel":172},"ch02","One giant value: outliers","Chapter 02","2 Outliers",{"id":174,"type":43,"markdown":175},"outlier-setup","Ten children in a class get this much pocket money per week (₹): 40, 40, 50, 50, 60, 60, 60, 70, 80, 90. The mean is ₹60, the median is ₹60 and the mode is ₹60. Very tidy.\n\nThen a new child, Rohan, joins the class. Rohan gets **₹400** a week.",{"id":177,"type":62,"prompt":178,"options":179,"explanation":188},"predict-outlier","After Rohan joins (₹400), roughly what happens to the mean and median?",[180,182,184,186],{"id":66,"label":181},"Mean ≈ ₹91, median still ₹60",{"id":69,"label":183},"Mean ≈ ₹91, median ≈ ₹91",{"id":72,"label":185},"Mean still ₹60, median ≈ ₹230",{"id":75,"label":187},"Both jump to about ₹230","**The mean jumps to about ₹91, but the median stays at ₹60.** The total is now 600 + 400 = 1,000 over 11 children: 1,000 ÷ 11 ≈ ₹90.9. Rohan's ₹400 is shared out across everyone in the fair-share picture, lifting each share by about ₹31. But in the ordered line of 11 children, the middle (6th) child still gets ₹60. Rohan simply stands at the far end of the line; it does not matter to the median whether he has ₹91 or ₹4,000.\n\nNotice: now **10 of the 11 children get less than the mean**. The mean no longer describes a typical child.",{"id":190,"type":47,"variant":191,"title":192,"markdown":193},"def-outlier","definition","Outlier","An **outlier** is a value that is much bigger or much smaller than the rest of the data. Outliers can be genuine (one child really does get ₹400) or mistakes (someone typed 400 instead of 40). Either way, check them before trusting a mean.",{"id":195,"type":114,"component":115,"componentVersion":5,"config":196,"objective":219,"textAlternative":220,"help":221},"lab-outlier",{"datasets":197,"valueRange":207,"step":208,"challenges":209},[198],{"label":199,"unit":200,"values":201},"Weekly pocket money of 11 children, including Rohan","₹",[202,202,203,203,204,204,204,147,132,205,206],40,50,60,90,400,{"min":131,"max":206},10,[210,212,215,217],{"measure":136,"target":204,"prompt":211},"Suppose Rohan's ₹400 was a typing mistake. Fix it so the mean is back to ₹60.",{"measure":136,"target":213,"prompt":214},100,"Make the mean ₹100 without touching Rohan. Which single change works?",{"measure":143,"target":147,"prompt":216},"Make the median ₹70. Does changing Rohan help?",{"measure":146,"target":203,"prompt":218},"Make the range ₹50 by changing only one child.","Drag the outlier and the other values to compare how strongly each moves the mean and the median.","A dot plot of 11 amounts: 40, 40, 50, 50, 60, 60, 60, 70, 80, 90, 400. With Rohan's ₹400 the mean is 1,000 ÷ 11 ≈ **₹90.9**, the median is **₹60**, the mode is **₹60** and the range is 400 − 40 = **₹360**.\n\nChallenges:\n\n1. **Mean 60:** total must be 60 × 11 = 660; change 400 to **60**.\n2. **Mean 100:** total must be 1,100, i.e. 100 more. Change one ₹40 child to ₹140.\n3. **Median 70:** the 6th ordered value must be 70. Changing Rohan does nothing, because he is already at the top. Raising one ₹60 child is not enough (the 6th value is still 60); raise two of the ₹60 children to ₹70.\n4. **Range 50:** change 400 to 90 (90 − 40 = 50).\n\nMoving Rohan all the way from 400 to 0 changes the mean by 400 ÷ 11 ≈ ₹36, yet the median stays at ₹60.",{"simplerExplanation":222},"The mean shares out everything, so one huge amount lifts everyone's share. The median just looks at the child in the middle of the line, who does not change.",{"id":224,"type":47,"variant":225,"title":226,"markdown":227},"outlier-aha","aha","Resistant and sensitive","Statisticians say the median is **resistant** (or robust) to outliers, and the mean is **sensitive** to them. Neither is \"better\": sometimes you *want* the outlier to count (total rainfall for water planning), sometimes you do not (describing a typical child's pocket money). Choosing wisely is the subject of the next layer.",{"id":229,"type":157,"itemId":230,"prompt":231,"check":232,"hints":244,"feedback":245},"pr-outlier","data-handling.investigate-outlier-which","The ages of people at a family lunch are 8, 10, 12, 35, 38, 40 and 92 (great-grandmother). Which is closer to \"most people's\" age: the mean or the median?",{"kind":233,"options":234,"correct":243},"choice",[235,237,239,241],{"id":66,"label":236},"The mean, about 33.6",{"id":69,"label":238},"The median, 35",{"id":72,"label":240},"They are equal",{"id":75,"label":242},"Neither makes sense for ages",[69],[],{"correct":246,"incorrect":247},"The median is 35 and the mean is 235 ÷ 7 ≈ 33.6. Here they are close, because the big 92 and the small children roughly balance. Try it without the 92: the mean drops to about 23.8 but the median only to 23.5.","Calculate both: sum = 235, mean ≈ 33.6; ordered middle value = 35.",{"id":249,"type":53,"title":250,"eyebrow":251,"navLabel":252},"ch03","Change every value at once","Chapter 03","3 Shift and stretch",{"id":254,"type":62,"prompt":255,"options":256,"explanation":265},"predict-add5","A teacher adds **5 bonus marks** to every child's test score. Before the bonus: mean 12, median 11, mode 10, range 9. What are the new values?",[257,259,261,263],{"id":66,"label":258},"Mean 17, median 16, mode 15, range 14",{"id":69,"label":260},"Mean 17, median 16, mode 15, range 9",{"id":72,"label":262},"Mean 17, median 11, mode 10, range 9",{"id":75,"label":264},"Only the mean changes","**Mean 17, median 16, mode 15, range 9.** Adding 5 to everyone slides the whole data set 5 places along the number line. Every \"centre\" slides with it: the fair share gains 5, the middle child is 5 higher, the most common score is 5 higher. But the **gap** between the top and bottom scores does not change, because both moved by 5. Shifting data changes the averages but not the spread.",{"id":267,"type":62,"prompt":268,"options":269,"explanation":278},"predict-double","Now every value in a data set is **doubled**. The mean was 6 and the range was 8. What happens?",[270,272,274,276],{"id":66,"label":271},"Mean 12, range 8",{"id":69,"label":273},"Mean 12, range 16",{"id":72,"label":275},"Mean 6, range 16",{"id":75,"label":277},"Mean 8, range 10","**Mean 12 and range 16.** Doubling stretches the number line: every value, and every gap between values, doubles. For 2, 4, 6, 8, 10 (mean 6, range 8), doubling gives 4, 8, 12, 16, 20: mean 60 ÷ 5 = 12 and range 20 − 4 = 16. The median (6 → 12) and mode double too.",{"id":280,"type":80,"caption":281,"columns":282,"rows":287},"table-shift","What shifting and stretching do (data 2, 4, 6, 8, 10)",[283,284,285,286,87],"Change","New data","Mean","Median",[288,293,297,301,306,310],[289,290,291,291,292],"None","2, 4, 6, 8, 10","6","8",[294,295,296,296,292],"Add 5 to each","7, 9, 11, 13, 15","11",[298,299,300,300,292],"Subtract 2 from each","0, 2, 4, 6, 8","4",[302,303,304,304,305],"Double each","4, 8, 12, 16, 20","12","16",[307,308,309,309,300],"Halve each","1, 2, 3, 4, 5","3",[311,312,313,313,314],"Multiply by 10","20, 40, 60, 80, 100","60","80",{"id":316,"type":47,"variant":317,"title":318,"markdown":319},"shift-use","example","Why this is useful: the assumed-mean shortcut","To find the mean of 97, 102, 99, 104 and 98 quickly, subtract 100 from each: −3, 2, −1, 4, −2. Their sum is 0, so their mean is 0. Add the 100 back: the mean is **100**. You shifted the data, found an easy mean, and shifted back. Cricket scorers and shopkeepers do this in their heads all the time.",{"id":321,"type":114,"component":322,"componentVersion":5,"config":323,"objective":391,"textAlternative":392},"lab-what-changes","sort-game",{"prompt":324,"bins":325,"items":334,"seconds":131},"Start with 2, 4, 6, 8, 10 (mean 6, median 6). Sort each change by what it does.",[326,327,329,331],{"id":136,"label":264},{"id":143,"label":328},"Only the median changes",{"id":330,"label":76},"both",{"id":332,"label":333},"neither","Neither changes",[335,339,343,347,351,355,359,363,367,371,375,379,383,387],{"id":336,"label":337,"bin":136,"why":338},"w1","Change the 10 to 20","Mean becomes 7; the middle value is still 6.",{"id":340,"label":341,"bin":136,"why":342},"w2","Change the 2 to 0","Mean drops to 5.6; the middle value is still 6.",{"id":344,"label":345,"bin":136,"why":346},"w3","Change the 10 to 100","Mean shoots up to 24; the median stays 6. An outlier!",{"id":348,"label":349,"bin":330,"why":350},"w4","Add 1 to every value","The whole data set shifts: mean and median both become 7.",{"id":352,"label":353,"bin":330,"why":354},"w5","Change the 6 to 7","Mean 6.2 and median 7: the middle value itself changed.",{"id":356,"label":357,"bin":330,"why":358},"w6","Double every value","Mean and median both double to 12.",{"id":360,"label":361,"bin":330,"why":362},"w7","Add a new value 30","Mean 10 and median (6 + 8) ÷ 2 = 7.",{"id":364,"label":365,"bin":330,"why":366},"w8","Remove the 10","Mean 20 ÷ 4 = 5 and median (4 + 6) ÷ 2 = 5.",{"id":368,"label":369,"bin":332,"why":370},"w9","Add a new value 6","A value equal to the mean keeps the mean 6; the ordered middle is still 6.",{"id":372,"label":373,"bin":332,"why":374},"w10","Write the values in a different order","Order of writing changes nothing: sum and middle are the same.",{"id":376,"label":377,"bin":332,"why":378},"w11","Change 4 to 5 and 8 to 7","The total stays 30 and the middle stays 6.",{"id":380,"label":381,"bin":332,"why":382},"w12","Remove the 6","Mean 24 ÷ 4 = 6 and median (4 + 8) ÷ 2 = 6. Surprise!",{"id":384,"label":385,"bin":143,"why":386},"w13","Change 6 to 7 and 10 to 9","The total stays 30 (mean 6), but the middle value is now 7.",{"id":388,"label":389,"bin":143,"why":390},"w14","Change 6 to 5 and 2 to 3","The total stays 30 (mean 6), but the middle value is now 5.","Predict whether each change to a small data set moves the mean, the median, both or neither.","A sorting game starting from 2, 4, 6, 8, 10 (mean 6, median 6), with 14 changes to sort.\n\n- **Only the mean changes:** change 10 → 20 (mean 7); change 2 → 0 (mean 5.6); change 10 → 100 (mean 24). Changing an end value never moves the middle.\n- **Only the median changes:** 6 → 7 and 10 → 9 (total still 30, middle now 7); 6 → 5 and 2 → 3 (middle now 5).\n- **Both change:** add 1 to every value; 6 → 7; double every value; add a new value 30 (mean 10, median 7); remove the 10 (mean 5, median 5).\n- **Neither changes:** add a new 6; reorder the values; 4 → 5 and 8 → 7; remove the 6 (mean 24 ÷ 4 = 6, median (4 + 8) ÷ 2 = 6).\n\nKey ideas: the mean follows the total; the median follows the middle position.",{"id":394,"type":157,"itemId":395,"prompt":396,"check":397,"hints":400,"feedback":401},"pr-shift","data-handling.investigate-shift-temp","A week of temperatures in Shimla has a mean of 12 °C and a range of 7 °C. A thermometer was reading 2 °C too low every day, so each value is corrected by adding 2. What is the corrected **range**?",{"kind":161,"answer":398,"tolerance":131,"unit":399},7,"°C",[],{"correct":402,"incorrect":403},"Yes: adding 2 to every value moves the maximum and minimum by 2, so their difference stays 7 °C. (The mean becomes 14 °C.)","Adding the same amount to every value shifts everything; does the gap between the top and bottom change?",{"id":405,"type":53,"title":406,"eyebrow":407,"navLabel":408},"ch04","Always, sometimes or never?","Chapter 04","4 Always true?",{"id":410,"type":43,"markdown":411},"asn-intro","Mathematicians love statements that are **always** true, because you can build on them. Here are some claims about averages. For each, the investigator's job is to find either a reason it must always hold, or a single **counterexample** that breaks it.",{"id":413,"type":80,"caption":414,"columns":415,"rows":419},"table-asn","Claims about averages, tested",[416,417,418],"Claim","Verdict","Evidence",[420,424,428,431,434,437,440,444,447],[421,422,423],"The mean lies between the smallest and largest values.","Always","A fair share cannot be more than the richest or less than the poorest has.",[425,426,427],"The mean is one of the data values.","Sometimes","3, 7, 4, 6, 5 → 5 (yes). 1, 2, 6 → 3 (no).",[429,426,430],"The median is one of the data values.","Always with an odd count; with an even count 2, 4 → 3 (no).",[432,426,433],"A data set has a mode.","5, 7, 9 has no mode.",[435,426,436],"Mean, median and mode are all equal.","4, 5, 5, 6 → all 5. But 1, 1, 7 → mean 3, median 1, mode 1.",[438,426,439],"The range is bigger than the mean.","1, 9 → range 8, mean 5 (yes). 50, 52 → range 2, mean 51 (no).",[441,442,443],"The range can be negative.","Never","Max − min, and max is never below min.",[445,422,446],"If all values are equal, the range is 0.","Max and min are the same number.",[448,426,449],"More than half the values are above the mean.","1, 1, 1, 9: mean 3, only one value above. 1, 9, 9, 9: mean 7, three above.",{"id":451,"type":62,"prompt":452,"options":453,"explanation":462},"predict-above","Can **every value but one** in a data set be below the mean?",[454,456,458,460],{"id":66,"label":455},"No, about half must be above the mean",{"id":69,"label":457},"Yes, if one value is very large",{"id":72,"label":459},"Only if there are exactly two values",{"id":75,"label":461},"Only if all values are equal","**Yes.** Take 1, 1, 1, 1, 100. The mean is 104 ÷ 5 = 20.8, and four of the five values are below it. One huge value can drag the mean above everyone else. (It can never be above **all** of them, though: the fair share cannot be more than the biggest value.) This is the pocket-money story again, and it is why \"most people earn less than the average income\" is true in almost every country.",{"id":464,"type":157,"itemId":465,"prompt":466,"check":467,"hints":473,"feedback":474},"pr-asn","data-handling.investigate-asn-median","Is this statement always, sometimes or never true? \"Adding a new value larger than every other value increases the median.\"",{"kind":233,"options":468,"correct":472},[469,470,471],{"id":66,"label":422},{"id":69,"label":426},{"id":72,"label":442},[69],[],{"correct":475,"incorrect":476},"Sometimes. For 1, 2, 3 (median 2), adding 10 gives 1, 2, 3, 10 (median 2.5): it increased. But for 5, 5, 5 (median 5), adding 10 gives 5, 5, 5, 10 (median 5): no change. The median can rise or stay the same, but never fall.","Try a few small examples, especially ones with repeated values in the middle.",{"id":478,"type":47,"variant":479,"title":480,"markdown":481},"counterexample","nuance","One counterexample is enough","To show a claim is **not always** true, one example that breaks it is enough. To show it **is always** true, examples are not enough, however many you try; you need a reason that works for every possible data set. \"A fair share cannot exceed the biggest share\" is such a reason. You will build more of these arguments in Deepen.",{"id":483,"type":53,"title":484,"eyebrow":485,"navLabel":486},"ch05","Build a data set to order","Chapter 05","5 Build to order",{"id":488,"type":43,"markdown":489},"build-intro","Now turn the problem around. Instead of finding the averages of given data, **design** data with averages you choose. This is how puzzle-setters, exam writers and data scientists testing their programs think.",{"id":491,"type":492,"title":493,"problem":494,"steps":495},"we-build","worked_example","Five numbers with mean 7, median 5, mode 4 and range 8","Find five whole numbers with mean 7, median 5, only one mode, 4, and range 8.",[496,497,498,499,500,501,502],"Call the ordered numbers a ≤ b ≤ c ≤ d ≤ e.","Median 5 means the middle one is c = 5.","Mode 4 means 4 appears more often than anything else. It must appear at least twice and, being less than 5, must sit before c. So a = b = 4.","Range 8 means e − a = 8, so e = 4 + 8 = 12.","Mean 7 means the total is 7 × 5 = 35. So d = 35 − (4 + 4 + 5 + 12) = 35 − 25 = 10.","Answer: **4, 4, 5, 10, 12**. Check: mean 35 ÷ 5 = 7 ✓, median 5 ✓, mode 4 (only repeat) ✓, range 12 − 4 = 8 ✓.","Try the same puzzle with mean **6** instead: d would be 30 − 25 = 5, giving 4, 4, 5, 5, 12, which has two modes. So that puzzle is impossible! Constraints can clash.",{"id":504,"type":114,"component":115,"componentVersion":5,"config":505,"objective":526,"textAlternative":527},"lab-build",{"datasets":506,"valueRange":512,"step":5,"challenges":514},[507],{"label":508,"values":509},"Five numbers to design",[133,510,511,125,208],4,6,{"min":131,"max":513},20,[515,517,519,522,524],{"measure":136,"target":398,"prompt":516},"Make the mean 7.",{"measure":143,"target":162,"prompt":518},"Keep the mean 7 and make the median 5.",{"measure":520,"target":510,"prompt":521},"mode","Keep mean 7 and median 5, and make 4 the only mode.",{"measure":146,"target":125,"prompt":523},"Keep all of that and make the range 8. (One answer: 4, 4, 5, 10, 12.)",{"measure":146,"target":131,"prompt":525},"Start again: make the range 0 while keeping the mean 7.","Design five numbers step by step to hit a target mean, median, mode and range together.","Five values start at 2, 4, 6, 8, 10 (mean 6, median 6, no mode, range 8). Each challenge adds a condition.\n\n1. **Mean 7:** the total must be 35, five more than now; e.g. change 10 to 15.\n2. **Median 5:** the middle ordered value must be 5, with the total still 35.\n3. **Mode 4 only:** two values must be 4, both below the median.\n4. **Range 8:** biggest − smallest = 8. One full solution is **4, 4, 5, 10, 12**.\n5. **Range 0 with mean 7:** every value must be 7: 7, 7, 7, 7, 7.\n\nThe lab checks each target separately, so check the earlier conditions yourself as you go.",{"id":529,"type":157,"itemId":530,"prompt":531,"check":532,"hints":534,"feedback":535},"pr-build","data-handling.investigate-build-missing","Four numbers have a mean of 9. Three of them are 5, 8 and 12. What is the fourth?",{"kind":161,"answer":533,"tolerance":131},11,[164],{"correct":536,"incorrect":537},"Correct: the total must be 9 × 4 = 36, and 5 + 8 + 12 = 25, so the fourth is 36 − 25 = 11.","Find the total first: mean × count = 9 × 4.",{"id":539,"type":53,"title":540,"eyebrow":541,"navLabel":542},"ch06","Monsoon figures put to the test","Chapter 06","6 Monsoon data",{"id":544,"type":43,"markdown":545},"rain-invest","Back to Mumbai's monthly rainfall from Discover (IMD normals for 1991–2020 at Mumbai (Santacruz), rounded, in mm): 0, 0, 0, 0, 7, 526, 920, 561, 384, 91, 11, 2. Total ≈ 2,502 mm.\n\nPredict before you calculate: will the mean month and the median month be close?",{"id":547,"type":62,"prompt":548,"options":549,"explanation":558},"predict-rain","For Mumbai's 12 monthly rainfall values, which is true?",[550,552,554,556],{"id":66,"label":551},"Mean and median are both about 200 mm",{"id":69,"label":553},"The mean is about 210 mm but the median is only about 9 mm",{"id":72,"label":555},"The median is much bigger than the mean",{"id":75,"label":557},"The mode is the best description","**The mean is about 209 mm but the median is only 9 mm.** Ordered, the values are 0, 0, 0, 0, 2, 7, 11, 91, 384, 526, 561, 920. With 12 values, the median is the mean of the 6th and 7th: (7 + 11) ÷ 2 = 9. More than half the months are almost completely dry, so the middle month is dry. The mean, however, shares out the huge monsoon totals over all 12 months. Neither alone tells the whole story: the data has two very different seasons.",{"id":560,"type":43,"markdown":561},"rain-days","Rainfall in millimetres covers a huge range (0 to 920), which is hard to plot. A friendlier measure is the number of **rainy days** in each month. For Mumbai (Santacruz) the IMD normals for 1991–2020, rounded to whole days, are: Jan 0, Feb 0, Mar 0, Apr 0, May 1, Jun 14, Jul 23, Aug 21, Sep 14, Oct 4, Nov 1, Dec 0. That makes about 78 rainy days a year, nearly all from June to September.",{"id":563,"type":114,"component":115,"componentVersion":5,"config":564,"objective":586,"textAlternative":587},"lab-rainy-days",{"datasets":565,"valueRange":573,"step":5,"challenges":575},[566],{"label":567,"unit":568,"values":569},"Mumbai (Santacruz): rainy days in each month, Jan to Dec","days",[131,131,131,131,5,570,571,572,570,510,5,131],14,23,21,{"min":131,"max":574},31,[576,579,581,583],{"measure":143,"target":577,"prompt":578},3,"Imagine a year with a longer monsoon. Change months so the median becomes 3 rainy days.",{"measure":136,"target":398,"prompt":580},"Make the mean 7 rainy days per month. How many extra rainy days is that in the year?",{"measure":520,"target":5,"prompt":582},"Make 1 day the only mode.",{"measure":146,"target":584,"prompt":585},25,"A record wet July: make the range 25 days.","Explore how a seasonal pattern gives a mean far above the median, and try changing the season.","A dot plot of 12 monthly counts: 0, 0, 0, 0, 1, 14, 23, 21, 14, 4, 1, 0. Sum 78; mean 78 ÷ 12 = **6.5**; ordered, the 6th and 7th values are 1 and 1, so the median is **1**; the mode is **0** (five months); the range is 23 − 0 = **23**.\n\nChallenges: for median 3, the 6th and 7th ordered values must average 3, e.g. change two dry months to 3 days each; for mean 7 the year needs 84 rainy days, 6 more; for mode 1 only, make more months equal to 1 than to 0, e.g. change two of the 0s to 1 (then 1 appears 4 times, 0 three times); for range 25, set July to 25.\n\nThe mean (6.5) is more than six times the median (1): a sign of a lopsided, **skewed** data set.",{"id":589,"type":47,"variant":48,"title":590,"markdown":591},"skew","Lopsided data","When a few very large values stretch the data out to one side, statisticians call it **skewed**. In skewed data the mean is pulled towards the long tail, away from the median. Rainfall, incomes, city populations and YouTube views are all strongly skewed. Heights of children in one class are not: they bunch symmetrically around the middle, so the mean and median are close.",{"id":593,"type":53,"title":594,"eyebrow":595,"navLabel":596},"ch07","Comparing two groups","Chapter 07","7 Comparing",{"id":598,"type":43,"markdown":599},"compare","The school team needs one more batter. Two players have the same mean over their last 8 innings:\n\n- **Arjun:** 40, 45, 50, 40, 45, 50, 40, 50\n- **Bhavna:** 0, 120, 5, 90, 0, 75, 60, 10\n\nBoth average 45 runs. So they are equally good? Not so fast. Look at the **spread**.",{"id":601,"type":80,"caption":602,"columns":603,"rows":607},"table-compare","Two batters, same mean",[604,605,606],"Measure","Arjun","Bhavna",[608,611,613,615,618],[609,610,610],"Total runs","360",[285,612,612],"45",[286,612,614],"(10 + 60) ÷ 2 = 35",[87,616,617],"50 − 40 = 10","120 − 0 = 120",[619,90,300],"Scores under 20",{"id":621,"type":114,"component":115,"componentVersion":5,"config":622,"objective":634,"textAlternative":635},"lab-batters",{"datasets":623,"valueRange":632,"step":162},[624,627],{"label":625,"unit":120,"values":626},"Arjun: last 8 innings",[202,18,203,202,18,203,202,203],{"label":628,"unit":120,"values":629},"Bhavna: last 8 innings",[131,630,162,205,131,631,204,208],120,75,{"min":131,"max":633},150,"Compare two batters with the same mean but very different spreads.","Two dot plots. **Arjun** (40, 45, 50, 40, 45, 50, 40, 50): all dots packed between 40 and 50. Mean 45, median 45, range 10. **Bhavna** (0, 120, 5, 90, 0, 75, 60, 10): dots scattered from 0 to 120. Mean 45, median 35, range 120.\n\nArjun is **consistent**: you can count on about 45. Bhavna is **explosive**: she might win a match on her own or be out for 0. Half her innings were 10 or less. Which to choose depends on the situation: a steady opener who protects the start of the innings, or a big hitter to chase a large total quickly. The mean alone cannot make that decision; you need the spread too.",{"id":637,"type":638,"prompt":639},"reflect-batters","reflection","You are the captain. Your team needs 180 runs from 20 overs and is 30 for 4. Would you rather send in Arjun or Bhavna next? Would your answer change if you only needed 60 runs to win with plenty of overs left? Use the words *mean*, *median* and *range*.",{"id":641,"type":53,"title":642,"eyebrow":643,"navLabel":644},"ch08","Missing-value detective","Chapter 08","8 Missing values",{"id":646,"type":43,"markdown":647},"missing-intro","A smudge of ink, a torn page, a forgotten entry: real data often has gaps. If you know an average, you can often recover the missing value. The key is always to switch from the mean to the **total**: sum = mean × count.",{"id":649,"type":492,"title":650,"problem":651,"steps":652},"we-missing-1","The torn attendance register","A class recorded attendance for 6 days: 32, 35, 30, ?, 34, 33. The mean attendance was 33. How many attended on day 4?",[653,654,655,656],"Total over 6 days = mean × count = 33 × 6 = 198.","Known days: 32 + 35 + 30 + 34 + 33 = 164.","Missing day = 198 − 164 = **34**.","Check: (32 + 35 + 30 + 34 + 34 + 33) ÷ 6 = 198 ÷ 6 = 33 ✓.",{"id":658,"type":492,"title":659,"problem":660,"steps":661},"we-missing-2","A value leaves the group","Seven friends have a mean height of 140 cm. One friend, 152 cm tall, moves to another city. What is the mean height of the remaining six?",[662,663,664,665],"Old total = 140 × 7 = 980 cm.","New total = 980 − 152 = 828 cm.","New mean = 828 ÷ 6 = **138 cm**.","It went down, because the friend who left was taller than the mean. Removing a value above the mean pulls the mean down.",{"id":667,"type":492,"title":668,"problem":669,"steps":670},"we-missing-median","A missing value and the median","Five numbers in order are 3, 7, x, 12, 15 and their median equals their mean. Find x.",[671,672,673,674],"The median is the middle value, x.","The mean is (3 + 7 + x + 12 + 15) ÷ 5 = (37 + x) ÷ 5.","Set them equal: x = (37 + x) ÷ 5, so 5x = 37 + x, 4x = 37, x = 9.25.","Check the order: 7 ≤ 9.25 ≤ 12 ✓. Mean = 46.25 ÷ 5 = 9.25 ✓. So **x = 9.25**.",{"id":676,"type":157,"itemId":677,"prompt":678,"check":679,"hints":683,"feedback":684},"pr-missing","data-handling.investigate-missing-kg","The mean mass of 5 watermelons is 4.2 kg. Four of them weigh 3.8, 4.5, 4.0 and 4.9 kg. What is the mass of the fifth?",{"kind":161,"answer":680,"tolerance":681,"unit":682},3.8,0.01,"kg",[],{"correct":685,"incorrect":686},"Yes: total = 4.2 × 5 = 21 kg; the four weigh 17.2 kg; 21 − 17.2 = 3.8 kg.","Total = 4.2 × 5 = 21 kg. Subtract the four known masses.",{"id":688,"type":157,"itemId":689,"prompt":690,"check":691,"hints":693,"feedback":695},"pr-missing-2","data-handling.investigate-missing-new","A class of 30 has a mean mark of 60. A new student joins and the mean becomes 61. What did the new student score?",{"kind":161,"answer":692,"tolerance":131},91,[694],"The new student must also lift everyone else's share by 1: 60 + 31 = 91.",{"correct":696,"incorrect":697},"Correct: old total 30 × 60 = 1,800; new total 31 × 61 = 1,891; the new student scored 1,891 − 1,800 = 91.","Compare the totals before and after: 30 × 60 and 31 × 61.",{"id":699,"type":53,"title":700,"eyebrow":701,"navLabel":702},"ch09","How many should you ask?","Chapter 09","9 Sample size",{"id":704,"type":43,"markdown":705},"sample-intro","In Discover, Class 6B's snack survey of all 30 children gave samosa as the clear winner (9 votes). But suppose the class monitor was in a hurry and stopped after asking only the first few children. Would she have got the same answer?\n\nThe survey answers were recorded in the order the children answered. Here is what the leader looked like as more and more answers came in.",{"id":707,"type":80,"caption":708,"columns":709,"rows":713},"table-sample","Class 6B snack survey: the counts after the first n answers",[710,711,712],"Answers so far","Counts","Mode so far (votes)",[714,718,721,725,729,733],[715,716,717],"5","Samosa 2, Idli 1, Fruit 1, Poha 0, Biscuits 1","Samosa (2)",[33,719,720],"Samosa 4, Idli 3, Fruit 1, Poha 0, Biscuits 2","Samosa (4)",[722,723,724],"15","Samosa 4, Idli 4, Fruit 4, Poha 0, Biscuits 3","Fruit and Idli and Samosa (4)",[726,727,728],"20","Samosa 5, Idli 4, Fruit 6, Poha 2, Biscuits 3","Fruit (6)",[730,731,732],"25","Samosa 7, Idli 5, Fruit 6, Poha 4, Biscuits 3","Samosa (7)",[96,734,735],"Samosa 9, Idli 7, Fruit 6, Poha 5, Biscuits 3","Samosa (9)",{"id":737,"type":62,"prompt":738,"options":739,"explanation":748},"predict-sample","The monitor stopped after the first 20 answers. Which snack would she have ordered?",[740,742,744,746],{"id":66,"label":741},"Samosa, the true favourite",{"id":69,"label":743},"Fruit",{"id":72,"label":745},"Idli",{"id":75,"label":747},"It would be a tie","**Fruit.** After 20 answers, fruit had 6 votes and samosa 5, because several fruit-lovers happened to answer early. The last 10 children included four more samosa votes and no more fruit votes, which flipped the result. After 15 answers it was a three-way tie. **Small samples can give the wrong answer just by chance**, and the order in which people are asked (who is standing near the front?) can add bias. The fix: ask everyone if you can; if you cannot, ask a large group chosen fairly, for example by drawing roll numbers from a box.",{"id":750,"type":47,"variant":479,"title":751,"markdown":752},"sample-nuance","Why big surveys ask only some people","The Census of India tries to count everyone, but it is enormous work and happens only about once a decade. Most national surveys, such as those on household spending or jobs, ask a carefully chosen **sample** of a few lakh households rather than every household in India — the 2011 Census counted about 25 crore of them, and the number has grown since. If the sample is chosen fairly and is large enough, its results are very close to what asking everyone would give. The Indian statistician P. C. Mahalanobis was a pioneer of such large-scale sample surveys in the 1930s–1950s.",{"id":754,"type":638,"prompt":755},"reflect-sample","You want to know the favourite sport of children in your whole school of 800. You can only ask 50. Describe how you would choose those 50 so the answer is fair. Who might you accidentally leave out if you just asked the first 50 children at the gate?",{"id":757,"type":53,"title":758,"eyebrow":759,"navLabel":760},"ch10","What we found","Chapter 10","10 Findings",{"id":762,"type":43,"markdown":763},"findings","Here is what our experiments showed, in the language of an investigator's report.\n\n- **Adding or removing a value** moves the mean towards or away from that value, by (distance from the mean) ÷ (new count). The median moves at most half a step along the ordered list.\n- **Outliers** drag the mean a long way but barely touch the median. The median is *resistant*, the mean *sensitive*.\n- **Adding a constant** to every value shifts mean, median and mode by that constant and leaves the range unchanged. **Multiplying** every value multiplies all of them, range included.\n- **Totals** are the key to missing values: sum = mean × count.\n- **Small samples** can crown the wrong winner by chance; ask many people, chosen fairly.\n- **Same average, different story:** two data sets can share a mean (or a median) and still be very different; always check the spread.",{"id":765,"type":766,"title":767,"terms":768},"glossary-investigate","glossary","Words from this layer",[769,773,776,779,783,787,791,794],{"term":770,"meaning":771,"example":772},"outlier","A value much larger or smaller than the rest of the data.","₹400 among amounts of ₹40–₹90.",{"term":774,"meaning":775},"resistant (robust)","Hardly affected by outliers. The median is resistant.",{"term":777,"meaning":778},"sensitive","Strongly affected by extreme values. The mean and the range are sensitive.",{"term":780,"meaning":781,"example":782},"skewed","Lopsided: a few very large (or very small) values stretch the data to one side, pulling the mean away from the median.","Monthly rainfall in Mumbai.",{"term":784,"meaning":785,"example":786},"symmetric","Balanced on both sides of the middle, so the mean and median are close.","Heights in one class.",{"term":788,"meaning":789,"example":790},"consistent","Having a small spread: values stay close together.","Arjun: 40 to 50 runs every time.",{"term":478,"meaning":792,"example":793},"One example that shows a general claim is false.","1, 2, 6 has mean 3, which is not in the data.",{"term":795,"meaning":796,"example":797},"assumed mean","A convenient number subtracted from every value to make a mean easier to find, then added back.","Use 100 for 97, 102, 99, 104, 98.",{"id":799,"type":800,"title":801,"questions":802},"quiz-investigate","quiz","Experiment check",[803,815,825,838,850,862,875,888,899],{"itemId":804,"prompt":805,"options":806,"correct":66,"why":814},"data-handling.investigate-q-add","The mean of 6 numbers is 10. A 7th number, 10, is added. The new mean is…",[807,808,810,812],{"id":66,"label":33},{"id":69,"label":809},"more than 10",{"id":72,"label":811},"less than 10",{"id":75,"label":813},"cannot tell","A value equal to the mean keeps the mean the same: 70 ÷ 7 = 10.",{"itemId":816,"prompt":817,"options":818,"correct":69,"why":824},"data-handling.investigate-q-outlier","Which is least affected by one very large value?",[819,820,821,822],{"id":66,"label":285},{"id":69,"label":286},{"id":72,"label":87},{"id":75,"label":823},"Total","The median only depends on the middle position; the mean, range and total all change.",{"itemId":826,"prompt":827,"options":828,"correct":72,"why":837},"data-handling.investigate-q-shift","Every value in a data set is increased by 3. The range…",[829,831,833,835],{"id":66,"label":830},"increases by 3",{"id":69,"label":832},"triples",{"id":72,"label":834},"stays the same",{"id":75,"label":836},"decreases by 3","Max and min both rise by 3, so their difference is unchanged.",{"itemId":839,"prompt":840,"options":841,"correct":72,"why":849},"data-handling.investigate-q-mult","Every value is multiplied by 4. If the median was 2.5, it becomes…",[842,844,846,847],{"id":66,"label":843},"2.5",{"id":69,"label":845},"6.5",{"id":72,"label":33},{"id":75,"label":848},"6.25","Multiplying every value multiplies the median too: 2.5 × 4 = 10.",{"itemId":851,"prompt":852,"options":853,"correct":66,"why":861},"data-handling.investigate-q-missing","Five numbers have mean 8. Four of them are 6, 7, 9 and 10. The fifth is…",[854,855,857,859],{"id":66,"label":292},{"id":69,"label":856},"7",{"id":72,"label":858},"40",{"id":75,"label":860},"32","Total 40; 6 + 7 + 9 + 10 = 32; 40 − 32 = 8.",{"itemId":863,"prompt":864,"options":865,"correct":66,"why":874},"data-handling.investigate-q-remove","Removing a value that is below the mean makes the mean…",[866,868,870,872],{"id":66,"label":867},"go up",{"id":69,"label":869},"go down",{"id":72,"label":871},"stay the same",{"id":75,"label":873},"become zero","Taking away a below-fair-share value leaves more per remaining person.",{"itemId":876,"prompt":877,"options":878,"correct":66,"why":887},"data-handling.investigate-q-skew","In a data set of incomes, the mean is ₹90,000 and the median ₹30,000. This suggests…",[879,881,883,885],{"id":66,"label":880},"a few very high incomes",{"id":69,"label":882},"a few very low incomes",{"id":72,"label":884},"everyone earns about the same",{"id":75,"label":886},"a calculation mistake","A mean far above the median signals a long tail of high values.",{"itemId":889,"prompt":890,"options":891,"correct":69,"why":898},"data-handling.investigate-q-asn","\"A data set with an even number of values has a median that is one of the values.\" This is…",[892,894,896],{"id":66,"label":893},"always true",{"id":69,"label":895},"sometimes true",{"id":72,"label":897},"never true","For 2, 4 the median is 3 (not a value). For 2, 5, 5, 9 the median is 5 (a value).",{"itemId":900,"prompt":901,"options":902,"correct":69,"why":911},"data-handling.investigate-q-same","Two classes have the same mean mark. What else do you need to know to say which is more consistent?",[903,905,907,909],{"id":66,"label":904},"The mode",{"id":69,"label":906},"A measure of spread, such as the range",{"id":72,"label":908},"The total",{"id":75,"label":910},"Nothing more","Consistency is about spread, not centre.",{"id":913,"type":914,"title":915,"points":916},"cheat-investigate","summary","Cheat sheet",[917,918,919,920,921,922,923,924,925],"New value **above** the mean → mean rises; **below** → falls; **equal** → unchanged. Change in mean = (new value − old mean) ÷ new count.","The median moves at most half a step when one value is added; an outlier hardly affects it.","**Outlier:** far from the rest. Check it; it may be real or a mistake. It drags the mean and stretches the range.","**Add k to every value:** mean, median, mode + k; range unchanged. **Multiply by k:** all of them × k, range too.","**Missing value:** use sum = mean × count, then subtract the known values.","**Skewed data** (rainfall, incomes): mean pulled towards the long tail, away from the median.","**Same mean ≠ same data.** Compare spreads (range) to judge consistency.","**Sample size:** a small or unfair sample can give the wrong mode or mean just by chance.","One counterexample disproves an \"always\"; a reason is needed to prove one.",{"id":927,"type":928,"conceptId":929,"relation":930,"explanation":931},"conn-patterns","connection","patterns","related_to","Adding the same number to every term of a sequence shifts its mean by that number, just like shifting data.",{"id":933,"type":928,"conceptId":934,"relation":935,"explanation":936},"conn-four","four-operations","helps_understand","Missing-value problems are inverse operations: multiply mean by count to undo the division.",{"id":938,"type":928,"conceptId":939,"relation":940,"explanation":941},"conn-elec","electricity","applied_in","Monthly electricity use is seasonal data; a hot summer is like an outlier that lifts the mean bill.",{"id":943,"type":944,"sourceIds":945},"sources-investigate","sources",[946,947,948,949,950,951,952],"data-handling-ncert-class7","data-handling-khan-summarizing","data-handling-mathsisfun-central","data-handling-imd","data-handling-imd-normals","data-handling-wiki-mahalanobis","data-handling-census-india",[946,947,948,949,950,951,952],"needs_review",{"generatedBy":956,"notes":957},"claude-code","Draft generated with Python-checked statistics; pending owner review.","431730cc21aa0ac8b88107ca21465af1e3991411f431dec2c5af4dda7237bd4b",{"component:data-lab@1":960,"logic:practice":961,"component:sort-game@1":962,"source:data-handling-census-india":963,"source:data-handling-imd":964,"source:data-handling-imd-normals":965,"source:data-handling-khan-summarizing":966,"source:data-handling-mathsisfun-central":967,"source:data-handling-ncert-class7":968,"source:data-handling-wiki-mahalanobis":969},"466896cc37735f48db03875fe9c9ce42fc8bcb7e5f937c9779d70513703b91bd","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","934b2e0d1b6222e6346f51e64ffc3e6c8d98db3be7d9a3c8d4cb3da843e8becc","834b555363edd0d5f3a42f5870d2acb5a33c3af5e40ecb43649732edd4cdce2c","f8d2cede5dff9df165f0ad49c28625d281417b1abac9405104cd7e3ff5c50f88","d3f02b5bb1750887d4c5e469441199469eba3f40ba38f08a0f15e6dc123789da","0e1e3ad9dc9f84bb51edeb70e657591671f3d8e7326c34f58c2de0c70e823635","73c4938c33c83e313b3e69ea6d4b3fcd121cfeab65eda7e07e1df7901cfaea2c","8b262d93bdc3277658143b72d0f370f2357f46f259c578ba699a36937d3759c6",{"state":971,"reviewer":972,"selfReview":973,"reviewedAt":974,"method":975},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597949]