Data Processing Class 12 Notes: Mean, Median and Mode

These Data Processing Class 12 notes compress Chapter 2 of the NCERT Practical Work in Geography Part-II textbook into one revision page: the meaning of the mean, median and mode, every formula with its symbols, stepwise worked computations and the comparison that ties the three averages together.

The notes follow the 2026-27 session and are built so you can revise from them alone, even the night before.

Chapter 1 left you with organised and presented data. This chapter is the next stage of practical geography: processing that data to extract a single representative number. The question behind the whole chapter is simple — which one value best stands for an entire set of observations?

The textbook names three families of statistical techniques but teaches only one. These notes mark that scope from the start, so your revision time goes only to what the chapter actually covers.

What Data Processing Covers in Class 12 Geography

Data processing begins where Chapter 1 ended: the data you organised and presented become the raw material for statistical analysis (NCERT, p. 13). The table below shows the three families of statistical techniques the chapter identifies, with the context each one serves.

Only the first family is taught in this chapter. Measures of central tendency are also known as statistical averages, and the chapter’s three members are the mean, the median and the mode (NCERT, p. 13). Each one is a different method of determining a single representative number, suited to different types of data sets (p. 14).

Measures of dispersion and measures of relationship appear only as context — no computations for them follow.

Mean — Direct and Indirect Methods for All Data Types

Geographic characteristics such as rainfall, elevation, population density, educational attainment and age groups vary from place to place (NCERT, p. 13). The mean is the value derived by summing all the values and dividing by the number of observations (p. 13). It can be calculated by the direct or the indirect method, for both ungrouped and grouped data (p. 14).

Ungrouped data — the direct method

Add every raw value exactly as it appears, then divide by the number of observations (NCERT, p. 14). The formula is:

\[ \bar{X} = \frac{\sum x}{N} \]

Symbols: \(\bar{X}\) = mean; \(\sum\) = sum of a series of measures; \(x\) = a raw score in the series; \(\sum x\) = the sum of all the measures; \(N\) = the number of measures (NCERT, p. 14).

Ungrouped data — the indirect method

When the number of observations is large, the indirect method is normally used (NCERT, p. 15). It reduces the values to smaller numbers by subtracting a constant — the assumed mean — from every value. This operation is known as coding (p. 15). The mean is then worked out from these reduced numbers instead of the raw ones.

\[ \bar{X} = A + \frac{\sum d}{N} \]

Symbols: \(A\) = the subtracted constant (assumed mean); \(\sum d\) = the sum of the coded scores; \(N\) = the number of individual observations (NCERT, p. 15). Because the same constant was removed from every value, adding it back at the end restores the true mean — the direct and indirect methods must give the same answer (p. 15).

Grouped data — the direct method

Once scores are grouped into a frequency distribution, the individual values lose their identity and each class is represented by its midpoint (NCERT, pp. 15-16). Multiply each midpoint by its frequency, add the products, and divide by the total number of observations \(N\).

\[ \bar{X} = \frac{\sum fx}{N} \]

Symbols: \(f\) = frequencies; \(x\) = midpoints of class intervals; \(N\) = number of observations, which may also be defined as \(\sum f\) (NCERT, p. 16).

Grouped data — the indirect method

The same coding idea is applied to the midpoints. The assumed mean is taken as the midpoint of a class chosen near the middle of the series, which keeps the deviations small (NCERT, p. 17).

\[ \bar{x} = A \pm \frac{\sum fd}{N} \]

Symbols: \(A\) = midpoint of the assumed mean group; \(f\) = frequency; \(d\) = deviation of each midpoint from \(A\); \(N\) = sum of cases or \(\sum f\); \(i\) = interval width (NCERT, p. 17).

The book’s procedure for the grouped indirect mean (NCERT, p. 17):

1. Assume the mean in a class as near to the middle of the series as possible.
2. List the deviation \(d\) of the midpoint of each class from \(A\).
3. Multiply each frequency by its deviation to get \(fd\).
4. Add the positive and negative values of \(fd\) separately and find their absolute difference — this is \(\sum fd\).
5. Restore the sign of \(\sum fd\) after \(A\) using the \(\pm\) in the formula, then divide by \(N\).

Edge case worth remembering: the indirect mean method works for both equal and unequal class intervals (NCERT, p. 17). Irregular class widths such as 10–20 followed by 30–50 cause no problem. And placing the assumed mean near the middle of the series keeps the computations smallest (p. 17).

Median — the Middle Rank Value, Ungrouped and Grouped

The median is a positional average — the point in a distribution with an equal number of cases on each side of it (NCERT, p. 18). It is expressed with the symbol \(M\) (p. 18). Because it depends on position rather than magnitude, the median is independent of the actual values of the series (p. 14).

Ungrouped data

Arrange the scores in ascending or descending order, then locate the central observation (NCERT, p. 14). The position is:

\[ \text{Median position} = \left( \frac{N+1}{2} \right)\text{th item} \]

For an even number of observations, the average of the two middle ranking values is the median (NCERT, p. 14). The textbook states this even-count rule but never demonstrates it; Worked Example 4 below does, so the rule is not left abstract.

Grouped data

For grouped data the median lies inside one class, located with cumulative frequencies:

\[ M = l + \frac{i}{f}\left( \frac{N}{2} – c \right) \]

Symbols: \(M\) = median; \(l\) = lower limit of the median class; \(i\) = interval; \(f\) = frequency of the median class; \(N\) = total number of observations; \(c\) = cumulative frequency of the pre-median class (NCERT, p. 19).

How to locate the median class (NCERT, p. 19):

1. Build a cumulative frequency column \(F\) by adding each class frequency to the running total.
2. Compute the median number \(N/2\).
3. Move down the \(F\) column until the value first greater than \(N/2\) is reached — that class is the median class.
4. Read off \(l\), \(i\) and \(f\) from the median class, and \(c\) from the class just above it.
5. Substitute these values into the formula.

Mode — the Most Frequently Occurring Value

The mode is the value that occurs most frequently in a distribution, symbolised as \(Z\) or \(M_0\) (NCERT, p. 20). The chapter notes that the mode is a measure less widely used compared with the mean and the median (p. 20).

Computing the mode for ungrouped data

Arrange all the measures in ascending or descending order, then identify the value with the highest count (NCERT, p. 20). That is the whole computation — the mode has no formula in this chapter.

Distribution types are defined by the number of tied values (NCERT, p. 20):

Scope note: the chapter computes the mode only for ungrouped data. No grouped-mode formula appears anywhere in it, so there is no grouped mode to compute.

Mean, Median and Mode in Normal and Skewed Distributions

The three measures are best compared with the help of the normal distribution curve. The normal curve is a bell-shaped frequency distribution in which many human traits — intelligence, personality scores, student achievements — are found (NCERT, p. 20).

It looks as it does because it is symmetrical: most observations lie on and around the middle value, and the number of observations reduces symmetrically as one approaches the extreme values (pp. 20-21).

Because a normal distribution is symmetrical, the mean, the median and the mode are the same score: the score with the highest frequency occurs in the middle, and exactly half of the scores occur above it and half below (NCERT, p. 21). Very high and very low scores occur rarely.

If the data are skewed or distorted, the three measures no longer coincide, and the effect of the skew has to be considered (p. 21).

The standard reading of the two skewed figures is:

• Positive skew (Fig. 2.2): the tail runs toward higher values; the order is \( \text{mode} \lt \text{median} \lt \text{mean} \).
• Negative skew (Fig. 2.3): the tail runs toward lower values; the order reverses to \( \text{mean} \lt \text{median} \lt \text{mode} \).

Fig. 2.1, Fig. 2.2 and Fig. 2.3 appear below with a guided reading in the distribution diagrams section.

The three averages compared

Keep the three apart with one line: the mean sums, the median ranks, the mode repeats.

Central Tendency Terms — a Definitions Table

Every term the chapter introduces, in one revision table.

Data Processing Formula Sheet with Symbols

The complete formula set for the chapter in one table. If any formula looks unfamiliar, check it directly against the chapter in the official NCERT textbook PDF, which opens the whole book so you can verify any symbol page by page.

Two book notes worth keeping: the indirect mean works for both equal and unequal class intervals (NCERT, p. 17); and in the grouped median formula \(f\) is the frequency of the median class, while \(c\) is the cumulative frequency of the pre-median class (p. 19) — mixing these two is the easiest error in the chapter.

Worked Examples — Every Method with Fresh Data

Six computations below cover every method in the chapter, each with original numbers so you can follow the full procedure with units, step by step. Nothing is skipped.

Worked Example 1 — Ungrouped mean by the direct method

This small imaginary data set is exactly the shape the chapter’s closing Activity suggests — an imaginary geographical example solved by the direct method from ungrouped data (NCERT, p. 22).

Data: annual rainfall of six districts, in mm: 145, 178, 162, 190, 154, 175.

Worked Example 2 — Ungrouped mean by the indirect method (same data)

Completes the Activity’s demand by solving the same imaginary data through the indirect method as well (NCERT, p. 22).

Worked Example 3 — Grouped mean, direct and indirect routes

Data: wheat yield in quintals per hectare (q/ha) for 35 sample plots.

Worked Example 4 — Ungrouped median: odd count and even count

The even-count case is the rule the textbook states on page 14 but never demonstrates — this is its demonstration.

Worked Example 5 — Grouped median

Data: daily wages of 40 workers, in rupees per day.

Worked Example 6 — Mode by inspection

Reading Fig. 2.1, 2.2 and 2.3 — the Distribution Diagrams

These three figures are the visual core of the comparison of the averages — and the support for exercise question 3(i), which asks for the relative positions of the averages in normal and skewed distributions with the help of diagrams. Here is the guided reading of each figure, in the order the chapter presents them.

Fig. 2.1 — Normal Distribution Curve

The figure shows the symmetric bell-shaped normal curve. Most observations lie on and around the middle value, so the highest-frequency score sits at the centre of the bell, and exactly half the scores lie on each side of it (NCERT, p. 21).

Because of this symmetry, the mean, the median and the mode are the same score — in the textbook’s example curve the central score is labelled 100 (p. 21). Very high and very low scores occur rarely, so they sit at the two thin ends of the bell.

Fig. 2.2 — Positive Skew

In a positive skew the distribution is distorted: the body of the data piles up toward the lower values and a longer tail stretches toward the higher values. The three averages no longer coincide (NCERT, p. 21).

In the standard reading of the figure, the mode lies at the peak of the curve, the median sits a little further along, and the mean is pulled furthest toward the tail by the few very high values — so the order is \( \text{mode} \lt \text{median} \lt \text{mean} \).

Fig. 2.3 — Negative Skew

The negative skew is the mirror case: the tail stretches toward the lower values while the mass of the data piles up toward the higher values. The three averages separate again (NCERT, p. 21), and the order reverses: \( \text{mean} \lt \text{median} \lt \text{mode} \). The mean is now the value dragged toward the low end of the tail.

Together, the three figures form the complete pattern: in a normal distribution all three averages are one score at the centre of the bell; in a positive skew they separate with the mode lowest and the mean highest; in a negative skew the order flips.

If you can sketch the bell, the right-hand tail and the left-hand tail, and attach the three averages in the correct order, you have the whole diagram-based answer ready.

Common Mistakes in Mean, Median and Mode (and the Corrections)

The errors below are the ones that actually appear in computation answers for this chapter. Each row gives the mistake, the correct rule, and a way to check yourself.

Answering the Data Processing Exercises with Complete Answers

The chapter closes with six questions and one activity on page 22. Everything needed is inside the chapter itself; below is the ingredient list each complete answer requires. Respect the word lengths the exercise states — about 30 words and about 125 words.

The two multiple-choice questions

Question 1(i): asks which measure of central tendency is not affected by extreme values. A complete answer chooses (d) Median, and the reasoning to carry is: the median is a positional average located by rank in the arranged series, so extreme values cannot move it (NCERT, p. 14).

Question 1(ii): asks for the measure that always coincides with the hump of any distribution. A complete answer chooses (d) Mode, and the reasoning is: the mode is the value of maximum frequency, and the highest-frequency score sits at the hump of a distribution (NCERT, pp. 20-21).

The about-30-words questions

Question 2(i) — define the mean.: A complete answer needs the definition (the value derived by summing all the values and dividing by the number of observations, NCERT p. 13), the formula stated in words (\(\bar{X}\) = sum of observations divided by the number of observations), and one short clause noting that it is the value about which the observations cluster (p. 13).

Question 2(ii) — advantages of using the mode.: A complete answer needs: the mode is identified by inspection after arranging the data (NCERT, p. 20); it gives the most typical, most repeated value; it involves no computation; and it should be balanced by the chapter’s own note that the mode is less widely used than the mean and the median (p. 20).

The about-125-words questions

Question 3(i) — relative positions in normal and skewed distributions, with diagrams.: A complete answer needs the three diagrams (the normal curve, a positive skew, a negative skew); the claim that in a normal distribution the mean, median and mode are the same score because the distribution is symmetrical — highest frequency at the centre, exactly half the scores on each side (NCERT, p. 21); and, for skewed distributions, their separation with the ordering fixed by the tail direction — mode below median below mean for a positive skew, the reverse for a negative skew. The figure walkthrough above rehearses this exact answer.

Question 3(ii) — applicability of the three measures from their merits and demerits.: A complete answer needs one paragraph per average: what each is based on (all values for the mean, rank for the median, frequency for the mode); the kind of geographic data each suits (district rainfall and factory wage groups for the mean, mountain peak heights for the median, student scores for the mode); and each measure’s limitation (the mean is dragged by extreme values, the median ignores the actual size of the values, the mode is less widely used and may be absent or multiple). Keep the three paragraphs tight enough for the 125-word cap.

The closing activity

The activity asks you to take an imaginary example applicable to geographical analysis and explain both the direct and indirect methods of calculating the mean from ungrouped data (NCERT, p. 22).

The complete response needs: a small imaginary geographic data set (six district rainfall values work well), a direct-method computation with \(\bar{X} = \sum x / N\), an indirect-method computation with \(\bar{X} = A + \sum d / N\), and a closing line noting that both give the same mean. Worked Examples 1 and 2 are this activity solved end to end.

Revision Summary — Mean, Median and Mode in One Glance

The whole chapter in one table.

Two-line recap: in a normal distribution all three averages coincide at the centre of the bell; in a skewed distribution they separate, and the ordering is fixed by the tail direction — \( \text{mode} \lt \text{median} \lt \text{mean} \) for a positive skew, the reverse for a negative skew.

If any method still feels shaky, the worked examples above model each route: Examples 1 and 2 the ungrouped mean, Example 3 the grouped mean, Example 4 the ungrouped median (both odd and even counts), Example 5 the grouped median, and Example 6 the mode.

Before closing, here is where these notes sit. The previous chapter’s Data – Its Source and Compilation notes cover how the raw data were organised for analysis, and the Graphical Representation of Data notes come right after this chapter in the same book.

Both chapters, along with this page, belong to the Class 12 Geography notes hub — and if you need notes for other subjects on the same level, browse the CBSE Class 12 notes section or the main CBSE notes library.

Frequently Asked Questions about Data Processing

Why is the median not affected by extreme values while the mean is?

Because the median is a positional average. It is located by rank in the arranged series, so it is independent of the actual values of the series (NCERT, p. 14). The mean, in contrast, adds every value into its sum, so a single extreme observation changes the total and drags the mean toward it.

The middle rank of the arranged series stays where it was, no matter how large the largest value becomes.

What is the difference between the direct and indirect method of calculating the mean?

The direct method adds the raw values and divides by the number of observations. The indirect method first codes the data — it subtracts an assumed constant \(A\) from every value to create small deviations \(d\) — and then adds the average deviation back: \( \bar{X} = A + \frac{\sum d}{N} \) (NCERT, p. 15).

Coding changes only the size of the numbers, not the average, so both methods give the identical mean; the indirect route is the convenient one when the values are large.

How do I identify the median class when computing a grouped median?

Compute \(N/2\), then move down the cumulative frequency column until you reach the first value greater than \(N/2\). The class that owns this cumulative frequency is the median class (NCERT, p. 19). Its lower limit gives \(l\), its frequency gives \(f\), and the cumulative frequency of the class just above it gives \(c\) for the formula.

Can a data set have more than one mode or no mode at all?

Yes, both are possible. Two values tying for the highest frequency make the set bimodal; three ties make it trimodal; many recurring values make it multimodal; and if no value repeats at all, it is without mode (NCERT, p. 20). A single mode exists only in a unimodal set.

When should I use the mean, median and mode in geography?

Use the mean when the data are fairly balanced and you want one overall representative value, such as the mean rainfall of a group of districts. Prefer the median when extreme values would distort the average — wage data with a few very high figures are the classic case.

Use the mode when you need the most common or typical value, such as the most frequently occurring peak height or temperature. These three uses echo the chapter’s contexts: rainfall and wages for the mean, mountain heights for the median, and test scores for the mode (NCERT, pp. 13-20).

Reference: NCERT Class 12 Practical Work in Geography Part-II textbook, chapter 2 Data Processing.