Organise, picture, summarise: how the methods work

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?

By 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.

Before you can summarise data, you need to know what kind of data it is, because that decides which tools work.

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.

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.

Numerical data comes in two flavours:

• 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.
• 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.

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.

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.

Class 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.

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.

A 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.

A bar graph is the workhorse of data display. Get these parts right and anyone can read yours at a glance:

• Title saying what the data is, where and when.
• Category axis (usually horizontal) with a label on each bar.
• Value axis (usually vertical) with a uniform scale that starts at 0, and a label saying the units.
• Bars of equal width with equal gaps; only the length of a bar carries information.

Bar graphs can stand up (vertical) or lie down (horizontal). Lying down is handy when category names are long, like the names of states.

The mean (also called the arithmetic mean) of a set of numbers is

mean = sum of all the values ÷ number of values.

Why 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.

The 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).

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.

• 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.
• 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.

So the median, like the mean, might not be one of the data values when n is even.

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).

Three special cases:

• 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.
• 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.
• 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".)

The range is the difference between the largest and smallest values:

range = maximum − minimum.

The 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.

Compare 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.

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.

• Mean: each row contributes value × frequency to the sum. Add a column f × x; the mean is (sum of f × x) ÷ (sum of f).
• Median: use a running total (cumulative frequency) to find which row contains the middle position.
• Mode: the row with the largest frequency.
• Range: the largest value with a non-zero frequency minus the smallest.