Types of data, and how to picture them

Every statistical question starts with a group you care about, and usually a group you cannot fully see. The population is everything you want to know about: all the households in a district, every firm in an industry, all the prices charged for onions on a given day. A sample is the part of it you actually observe.

A number that describes the population is a parameter. A number computed from the sample is a statistic, and it is your estimate of the parameter. They have different symbols, and CUET 2026 asks you to match them:

Two things follow. A parameter is a fixed number you don’t know. A statistic is a number you can compute, but it would come out differently if you drew a different sample. Much of statistics is about how far a statistic can be trusted, and Lesson 4 shows the first consequence: why a sample’s variance is calculated differently from a population’s.

This module’s running example is Ashanagar, an invented district. Its sixty households are treated as the whole population unless a lesson says otherwise, and every figure is illustrative. None of it is official data.

A variable is any characteristic that differs from one unit to the next. There are two families.

• Categorical (qualitative) variables put each unit in a group: religion, state, sector of employment. CUET 2018 asks which of age, annual income, grade point average and religion is categorical. Religion is.
• Numerical (quantitative) variables are measured or counted: age, income, a grade point average.

Numerical variables split again. A discrete variable takes separate values, usually counts: the number of members in a family (Hyderabad 2017). A continuous variable can take any value in a range, because it is measured: height, time, income. A quick test is to ask whether 2.7 is a possible value. For household size it is not. For time it is.

The kind of variable tells you what you are counting. The scale of measurement tells you which arithmetic is allowed. There are four, each adding a power to the one before.

Try sorting eight variables. The reason appears under each answer.

Cumulative percentages are the way into the opening question. A cumulative percentage says “this share of the data is at or below this category”, and that only makes sense if the categories can be put in order. We return to it at the end.

Economic data comes in three layouts, and the layout decides what questions you can ask of it.

CUET 2023 gives the figures for one state’s domestic product from 1990 to 2010 and asks what kind of dataset it is: a time series. The order of a time series matters, because yesterday’s value affects today’s, and that is why time series need their own methods. A cross-section has no order. A panel lets you ask something neither of the others can: how did this household change?

Stocks and flows

A second distinction runs through all of economics. A stock is measured at a moment: the money supply on 31 March, the capital in a factory, a country’s debt. A flow is measured over a period: income, investment, exports, GDP. Think of a bath. The water in it at any instant is the stock; the water running through the tap each minute is the flow. The change in the stock over a period is the inflow minus the outflow.

Hyderabad asked the same question in 2010, 2011 and 2015: which of GDP, money supply, exports and imports is a stock? The money supply. The other three are flows, and each is quoted per period.

Twenty numbers in a list tell you very little. Grouping them into classes and counting how many fall in each shows the pattern at a glance. The terms:

• Class interval: a range such as 400 to 450. Its limits are the two ends and its width, h, is the difference.
• Frequency, f: how many observations fall in the class. The relative frequency is f divided by the total, usually as a percentage.
• Class mark: the midpoint of the class, the single number that stands for it in later calculations.
• Cumulative frequency: a running total. The less-than version counts everything below the upper limit; the more-than version counts everything above the lower limit.

One rule trips people up. In an exclusive classification the classes read 400 to 450, 450 to 500, and a value of exactly 450 goes in the upper class. Every value belongs to exactly one class. Open-ended classes such as “above 600” are allowed, but they have no class mark, which is one reason the median and quartile deviation are preferred for such data (Lesson 4).

How many classes? Too few and the shape is flattened away; too many and each class holds a handful of values. A common starting point is Sturges’ rule, k ≈ 1 + 3.32 log₁₀ n, which gives about 5 classes for 20 observations and 7 for 60. It is only a starting point. The next section lets you see the effect.

When the classes are not the same width

Sometimes the classes are unequal: the data may be sparse at the top, so you use wider classes there. Then the frequency is no longer a fair measure of how crowded a class is, because a wide class collects more simply by being wide. The fair measure is the frequency density:

Suppose classes 0 to 10, 10 to 20, 20 to 40 and 40 to 80 hold 5, 15, 20 and 20 households. Drawing bars of height 5, 15, 20, 20 would make the wide classes look crowded. The densities are 0.5, 1.5, 1.0 and 0.5 households per rupee-thousand, and bars drawn to those heights have areas equal to the frequencies, which is the whole point of a histogram.

Here are the sixty Ashanagar incomes. Drag the class width and the table and the picture rebuild together. Then switch the picture: all three are drawn from the same table.

What each picture is for:

• A histogram shows the shape of a continuous variable. The bars touch, because there is no gap between one class and the next, and the area of each bar stands for its frequency. A bar chart is for categories; its bars are separated and only the height counts.
• A frequency polygon joins the class marks with straight lines, closed to zero at both ends. It is the histogram’s outline, and the easiest way to put two distributions on one chart.
• An ogive is a graph of cumulative frequency against the class limits. It answers “how many are below this value?” and, read the other way, “below what value is half the data?” The less-than and more-than ogives cross at the median. Lesson 2 uses it to read off quartiles.
• A pie chart shows shares of a whole, and a line chart is the natural picture of a time series. A scatter plot, for two variables, comes in Lesson 6.

JAM 2025 asks which of a histogram, a pictogram, an ogive and a frequency polygon is distinct from the others. The pictogram: it is a drawing with little icons, while the other three are all pictures of a frequency distribution.

Every formula in the next lessons adds up a column of numbers, and statisticians write that with the Greek capital sigma. ${\sum }_{i=1}^n{x}_i$ means “add x₁, x₂, and so on up to x_n”. The mean is then $\bar{x}=\frac{1}{n}\sum x_i$. Three rules cover almost everything:

One thing the rules do not say: $\sum x_i^2$ is not ${\left(\sum x_i\right)}^2$. For 1, 2 and 3 the first is 1 + 4 + 9 = 14 and the second is 6² = 36. That mistake is behind many wrong answers in the variance lesson.