Measures of Central Tendency Class 11 Notes

These are the revision notes for Measures of Central Tendency, Chapter 5 of the NCERT Class 11 Statistics for Economics textbook, prepared for the 2026-27 session. The chapter’s aim is simple to state: after tabular and graphic representation come the measures of central tendency, a numerical way to explain the data in brief.

This page gives you the three main averages — arithmetic mean, median and mode — with every formula, a consolidated formula sheet, five fully solved examples, the textbook’s activities worked out, and the mistakes students make most often.

Everything here is mapped to the chapter’s own objectives: understand the need for summarising data by a single number, distinguish between the types of averages, compute them, draw conclusions from them, and choose the right average for the situation.

These notes are part of the Class 11 Economics notes library, which includes an earlier chapter on the organisation of data and later chapters such as correlation; the wider Class 11 notes and the CBSE notes collection are also on this site. Reference: NCERT Class 11 Statistics for Economics textbook, chapter Measures of Central Tendency.

If any formula here looks unfamiliar, download the official chapter PDF hosted by NCERT to verify it against the printed book.

Why One Number Can Describe a Whole Data Set

The previous chapter showed how to present data in tables and graphs. This chapter adds the numerical method: it compresses an entire data set into one single representative value. Everyday examples are all around — average marks in a test, average rainfall in an area, average production in a factory, average income of a locality.

A single number is easier to compare, quote and act on than fifty separate observations.

The opening example sets up the whole chapter in miniature. Baiju, a farmer in Balapur village in Bihar’s Buxar district, owns 1 acre of land. To judge his economic condition, you compare him with the other 49 small farmers of the village. Three different comparisons give three different answers:

• Is his holding above the ordinary average? → the arithmetic mean answers this (NCERT, p. 59).
• Is it above what half the farmers own? → the median answers this.
• Is it above what most farmers own? → the mode answers this.

The chapter lists five things you should be able to do after studying it:

• understand the need for summarising a set of data by one single number;
• recognise and distinguish between the different types of averages;
• learn to compute the different types of averages;
• draw meaningful conclusions from a set of data;
• develop an understanding of which average is most useful in a particular situation.

The three averages covered in detail are arithmetic mean, median and mode. The book also names two further averages — geometric mean and harmonic mean — which suit certain situations, but their study is left for later (NCERT, p. 59).

The Three Averages: Which Question Does Each Answer?

Before any arithmetic, fix the big picture: the three averages are answers to three different questions, and choosing the wrong one gives a misleading answer.

Map it back to Baiju: “above average” is the mean’s question, “above half the farmers” is the median’s, and “above most farmers” is the mode’s. Keep this mapping in mind while solving and every formula below makes sense.

Key Terms in Central Tendency: Definitions With Examples

The chapter repeats a few terms constantly. Keep this table beside you while solving — every term, defined in plain words with a one-line example.

Arithmetic Mean: Definition, Formula and Its Two Properties

Arithmetic mean is the sum of the values of all observations divided by the number of observations (NCERT, p. 59). It is the most commonly used measure of central tendency and is denoted by \(\bar{X}\), read as “X bar”.

\[ \bar{X} = \frac{\Sigma X}{N} \]

where \(X\) = each observation, \(\Sigma\) (sigma) = sum of all observations, and \(N\) = total number of observations.

Why the mean deserves to represent the data: it is the one value that keeps the total of the data unchanged. If every observation were replaced by the mean, the sum of the data would stay exactly the same. That is why the mean works as an “overall level” — it spreads the total evenly across all observations.

The chapter highlights two properties of the arithmetic mean (NCERT, p. 63):

• The sum of deviations about the mean is zero: \(\Sigma (X – \bar{X}) = 0\). Positive deviations (values above the mean) and negative deviations (values below) cancel each other exactly. This is another way of saying the mean is the balancing point of the data.
• The mean is affected by extreme values: one very large or very small value pulls the mean towards it. This weakness is exactly why the median exists as an alternative — you will meet it in the next sections.

Three Ways to Compute the Mean of Individual Series

For ungrouped (“individual”) data, the book offers three methods. They are not three different means — all three give the same answer. The variety exists only to save arithmetic (NCERT, pp. 60–61):

Assumed mean logic: pick any convenient value \(A\) (a centrally located value works best), find each observation’s deviation \(d = X – A\), average those deviations, and add the average back to \(A\). You shift the origin to \(A\), then undo the shift. Any value — present in the data or not — can serve as the assumed mean.

Step deviation logic: if the deviations are still large, divide each by a common factor \(c\), average the shrunk values, then scale back up by multiplying by \(c\) at the end. The step deviation formula for individual data is \(\bar{X} = A + \frac{\Sigma d’}{N} \times c\), where \(d’ = \frac{X – A}{c}\) (NCERT, p. 61).

Mean for Discrete and Continuous Series: Frequencies and Mid-values

When data arrives grouped, the mean formula changes in only one respect: each value must be counted as many times as it occurs.

Discrete series (NCERT, p. 62): the data gives each value \(X\) together with its frequency \(f\). Multiply each value by its frequency, sum the products, and divide by the total frequency:

\[ \bar{X} = \frac{\Sigma fX}{\Sigma f} \]

Continuous series (NCERT, p. 63): values fall inside class intervals, so the actual observations inside each class are unknown. Each class is therefore represented by its mid-value \(m\) — the midpoint of the interval — and the same calculation follows:

\[ \bar{X} = \frac{\Sigma fm}{\Sigma f} \]

The mid-value of the class 20–30 is \(m = \frac{20 + 30}{2} = 25\). Class intervals may be exclusive (0–10, 10–20), inclusive (0–9, 10–19) or unequal (0–20, 20–50) — the mean is computed the same way in every case: take mid-values, multiply by frequencies, sum, and divide.

Both grouped cases also accept the assumed-mean and step-deviation shortcuts. Discrete: \(\bar{X} = A + \frac{\Sigma fd}{\Sigma f}\) (assumed mean) and \(\bar{X} = A + \frac{\Sigma fd’}{\Sigma f} \times c\) (step deviation). Continuous: the same formulas, but with \(m\) in place of \(X\) (NCERT, pp. 62–63).

Weighted Arithmetic Mean: When Every Item Doesn’t Count Equally

A simple mean treats every observation as equally important. When items differ in importance, the simple mean misleads (NCERT, pp. 63–64).

Suppose a family buys rice worth Rs 300 and cooking oil worth Rs 80 in a week. A simple average gives the two commodities equal importance — yet rice dominates the family’s budget, so a rise in rice prices hurts far more than a rise in oil prices.

Weights fix this: each item’s weight \(W\) (its importance, for example its share in the budget) decides how much it contributes. The formula is:

\[ \bar{X}_w = \frac{\Sigma WX}{\Sigma W} \]

For two items the book writes it as \(\frac{W_1 X_1 + W_2 X_2 + \dots + W_n X_n}{W_1 + W_2 + \dots + W_n}\). The same idea returns in Chapter 8 of the book, where index numbers weight price movements by their importance in the consumer’s budget.

Median: The Value That Splits the Data in Half

Median is the positional value that divides the distribution into two equal parts — 50% of observations lie on either side (NCERT, p. 64). It is the “middle” element of the data arranged in order of magnitude. Build it in this order:

1. Sort first. Arrange the data in ascending order. This step is compulsory — a position in unordered data means nothing.
2. Odd number of observations: the median is the middle value. For the list 5, 7, 6, 1, 8, 10, 12, 4, 3, the sorted values are 1, 3, 4, 5, 6, 7, 8, 10, 12 and the median is 6 (NCERT, p. 64).
3. Even number of observations: two values share the middle and the median is their arithmetic mean — for the two middle values 45 and 46, the median is 45.5 (NCERT, p. 65).
4. Position formula: the formula gives the position, not the value: \(\text{Median} = \text{size of the } \frac{(N+1)}{2}\text{-th item}\) (NCERT, p. 65).
5. Discrete series: locate the \(\frac{(N+1)}{2}\)-th item through the cumulative frequency column — the value in that row is the median (NCERT, p. 65).
6. Continuous series: first find the median class — the class in which the \(\frac{N}{2}\)-th item lies. Note this is \(\frac{N}{2}\), deliberately different from \(\frac{N+1}{2}\). Then interpolate with the formula below (NCERT, p. 66).

\[ \text{Median} = L + \frac{(N/2 – cf)}{f} \times h \]

Why the formula works: \(cf\) observations already lie below the median class. The median is the \(N/2\)-th item, so \((N/2 – cf)\) more items are needed inside the class.

Assuming they spread evenly across the width \(h\), the term \(\frac{(N/2 – cf)}{f} \times h\) converts that fraction of items into a distance, which is added to the lower limit \(L\).

Two practical notes: if class widths are unequal, no adjustment is required (NCERT, p. 66); and the median is not sensitive to all the values in the series — it concentrates on the central items (NCERT, p. 67). That is its defence against extreme values, connecting back to the mean’s weakness.

Quartiles and Percentiles: Quarters and Hundredths

Quartiles divide the data into four equal parts, each containing an equal number of observations (NCERT, p. 67).

• \(Q_1\) (lower quartile): 25% of items lie below it, 75% above.
• \(Q_2\): the second quartile — identical to the median.
• \(Q_3\) (upper quartile): 75% of items lie below it, 25% above.
• The central 50% of the data lies between \(Q_1\) and \(Q_3\).

The location method is the same as the median’s (NCERT, p. 68):

\[ Q_1 = \text{size of the } \frac{(N+1)}{4}\text{-th item} \qquad Q_3 = \text{size of the } \frac{3(N+1)}{4}\text{-th item} \]

Percentiles split the distribution into 100 equal parts, giving 99 dividing positions \(P_1, P_2, \dots, P_{99}\), with \(P_{50}\) equal to the median (NCERT, p. 67).

Understanding an 82nd percentile: the book phrases it as your position being “below 18 per cent” of candidates — that is, you sit below the top 18%. In plainer words, about 82% of candidates scored below you and about 18% scored above you. If one lakh candidates appeared, that is about 82,000 candidates below you, placing your rank near the 18,001st position from the top.

Mode: The Most Frequently Observed Value

Mode is the value that occurs most frequently in the data. The word comes from the French “la Mode” — the fashionable value — because it is repeated the highest number of times in the series (NCERT, p. 68). Everyday uses: a shoe manufacturer wants the shoe size with maximum demand; a shirt maker wants the style demanded most frequently.

The mode is denoted by \(M_o\).

Discrete series: read the mode by inspection — the value with the largest frequency. For the data 1, 2, 3, 4, 4, 5 the mode is 4, since 4 occurs twice. If the values 10, 20, 30, 40, 50 carry frequencies 2, 8, 20, 10, 5, the mode is 30, and because the mode is unique the data is unimodal. The mode need not be unique (NCERT, p. 68):

• bimodal data: two modes;
• multimodal data: more than two modes;
• no mode at all: the series 1, 1, 2, 2, 3, 3, 4, 4 has no mode because no value beats the others.

Continuous series: identify the modal class — the class with the largest frequency — then apply the formula (NCERT, p. 69):

\[ M_o = L + \frac{D_1}{D_1 + D_2} \times h \]

• \(L\) = lower limit of the modal class;
• \(D_1\) = difference between the modal class frequency and the frequency of the class preceding it (ignoring signs);
• \(D_2\) = difference between the modal class frequency and the frequency of the class succeeding it (ignoring signs);
• \(h\) = class interval of the distribution.

Why the formula leans: if the modal class were perfectly symmetric, the mode would sit at its centre. The \(D_1\) and \(D_2\) terms measure how different the two neighbouring classes are, and the mode is pulled toward the neighbour whose frequency is closer to the modal class’s own.

In Example 4 below, the mode lands near the lower end of the modal class because the preceding class (frequency 15) is far closer to the modal class (21) than the succeeding class (5) is.

Preconditions for the formula: class intervals must be equal and the series must be exclusive. If mid-points are given, reconstruct the class intervals first; if a “less than” cumulative table is given, convert it into an ordinary exclusive frequency table before finding the modal class (NCERT, p. 69).

Mean, Median and Mode: How Their Positions Compare

The chapter gives one compact fact about the three averages’ relative size (NCERT, p. 70). Writing arithmetic mean as \(M_e\), median as \(M_i\) and mode as \(M_o\), the possible orders are:

\[ M_e \gt M_i \gt M_o \quad \text{or} \quad M_e \lt M_i \lt M_o \]

In both cases the median sits between the mean and the mode. The chapter’s clue for remembering: the subscripts \(e, i, o\) occur in alphabetical order. So when you list mean, median, mode in that order, their magnitudes follow the same direction — all “greater than” or all “less than”.

Memory device: the alphabet carries the triple’s order for you — mean, median, mode walked in and lined up alphabetically by subscript.

Aid beyond the chapter (a check, not a requirement): the reason these orders exist is skewness. When a few very large values pull the distribution to the right, the mean is dragged up most, so mean > median > mode. When a few very small values pull it left, the order reverses.

The median always stays in the middle precisely because it ignores extremes.

Formula Sheet: Every Average on One Page

This is the page to revise from the night before. Every formula the chapter uses, in one table — each verified against the textbook pages listed in the sections above.

Symbols used by every formula above:

Units: every formula is unit-agnostic. The data’s unit — Rs, marks, acres, kg/hectare — flows straight into the final answer, so always write it there.

Solved Examples: Mean, Median, Mode and Quartiles Step by Step

Each example names the method first, then substitutes every value step by step with units on the final answer. Work each one through on paper — that is how the calculation pattern sticks.

Example 1 — Individual Series Mean by Direct and Step Deviation

Example 2 — Discrete Series Mean With Frequencies

Example 3 — Median of a Continuous Series

Example 4 — Mode of a Continuous Series

Example 5 — Lower and Upper Quartiles With Interpolation

The NCERT Activities, Answered

These are the textbook’s own open-ended activities, worked out. Use them to see each concept in action before attempting the exercises.

Activity 1 — Does \(\Sigma (X – \bar{X})\) really equal zero?

Activity 2 — Adding 2 to every observation

If every observation increases by 2, the whole data set shifts up by 2, so the mean shifts from 8 to 10. The mean moves with the data: add the same constant to every value and the mean rises by that same constant.

Activity 3 — First three items rise by 2; keep the mean at 8

Activity 4 — Replace 12 by 96

Activity 5 — The four-series table: mean vs median

Fill in Table 5.4 of the book:

The pattern: as the largest value explodes from 3 to 3000, the mean explodes too — 2, 11, 101, 1001 — while the median sits fixed at 2. Those outlying values are exactly the extreme values that leave the median untouched.

The activity then asks whether the median is a better method than the mean: for data of this kind, yes, because the median is not affected by extreme values.

Activity 6 — Which average for the shoe company?

A company making adult shoes wants the most popular size of shoes. That is the mode — the size demanded most frequently. The same logic drives the chapter’s examples of shirt styles and the “most fashionable” value: no arithmetic table is needed, just the largest frequency (NCERT, p. 68).

Common Mistakes Students Make With Averages

These are the chapter-specific traps, written the way students actually fall into them. Each row tells you the correction and why it matters.

How to Write Complete Answers for Computation Questions

Every average computation can follow the same repeatable structure:

1. Name the measure and write its formula — for example, “Median for continuous series: \(\text{Median} = L + \frac{(N/2 – cf)}{f} \times h\).”
2. Build or complete the working table — cumulative frequency, \(fX\), \(fd’\), mid-values, whatever the formula needs.
3. Substitute the actual numbers — write the substitution line, not just the final result.
4. Compute and state the unit — Rs, marks, acres, kg/hectare — on the final value.
5. Add one interpretation sentence — “about half the workers earn Rs 147.78 or less” — because this turns arithmetic into a conclusion.

“Which average is suitable?” — a decision rule. Use the purpose of the analysis and the nature of the distribution. The chapter’s conclusion is the map (NCERT, pp. 70–71):

Missing-frequency questions. The textbook’s exercises include a data set with one frequency unknown. The strategy is always algebraic: write the unknown frequency as \(x\), express \(\Sigma fX\) in terms of \(x\), set the given mean equal to \(\frac{\Sigma fX}{\Sigma f}\), and solve for \(x\). Then use the completed table to find the median with the \(N/2\) rule.

Apply this method to your own data — the worked values of the book’s exercise are printed with it.

60-Second Recap: A Five-Step Path to Any Average

Run every question through these five decisions when revising at the last minute:

1. Identify the series type: individual, discrete or continuous.
2. Decide the measure: mean for an overall level, median for the middle value, mode for the most frequent value.
3. Choose the fastest method: direct for small data; assumed mean or step deviation for many observations or large figures.
4. Apply the matching formula from the formula sheet.
5. Write the unit and an interpretation sentence.

Final reminders, one line each:

• Mean — sum of all values ÷ number of values (\(\frac{\Sigma X}{N}\)); weakness: one extreme value can drag it anywhere.
• Median — the middle value of sorted data; weakness: position only — the actual values hardly matter.
• Mode — the most frequent value; weakness: it may not exist, or there may be two or more.

And the memory device that locks the order: the subscripts of the three averages — \(M_e\) (mean), \(M_i\) (median), \(M_o\) (mode) — are in alphabetical order, so the magnitudes follow the alphabet: the three averages rise together or fall together, with the median always in the middle.

Frequently Asked Questions About Averages and Their Positions

Why do we use (N+1)/2 for the median position in individual and discrete series but N/2 in continuous series?

The two formulas locate different things. In individual and discrete series, \((N+1)/2\) gives the median’s position: an actual item when N is odd, or the mean of the two middle items when N is even.

In continuous series there are no individual items — \(N/2\) simply identifies which class interval contains the middle item, and the interpolation formula \(L + \frac{(N/2 – cf)}{f} \times h\) then places the median inside that class.

When is the median a better average than the mean in economics data?

Whenever extreme values are present or the distribution is open-ended. One very large income inflates the mean of a wage series — the Table 5.4 pattern shows a mean of 1001 against a median of 2 — while the median ignores the extremes, and unlike the mean it can still be computed for open-ended classes.

What is the difference between the assumed mean method and the step deviation method?

Assumed mean uses deviations \(d = X – A\) and the formula \(\bar{X} = A + \frac{\Sigma d}{N}\). Step deviation goes one step further: it divides each deviation by a common factor \(c\) to get \(d’\), then multiplies the averaged result back by \(c\), giving \(\bar{X} = A + \frac{\Sigma d’}{N} \times c\).

Step deviation only lightens arithmetic for large figures — the answer is identical.

Why does the mode formula need equal and exclusive class intervals?

Because it compares the modal class with its two neighbours using \(D_1\) and \(D_2\), on the assumption that every class has the same width \(h\) and continuous boundaries. Unequal or inclusive intervals break those comparisons; that is why a “less than” cumulative table must first be converted into an ordinary exclusive frequency table.

How are quartiles and percentiles related to the median?

The median is \(Q_2\), the second quartile, and also \(P_{50}\) among percentiles. Quartiles split the data into four equal parts (\(Q_1, Q_2, Q_3\)); percentiles split it into a hundred (\(P_1\) to \(P_{99}\)). \(Q_1\) and \(Q_3\) matter most as the walls of the central 50% of the data — 25% of items lie below \(Q_1\) and 25% above \(Q_3\).

What happens to the mean when one observation is replaced by a very large value?

It jumps. For 4, 6, 8, 10, 12 the mean is 8; replace 12 with 96 and the mean becomes 24.8. Because the mean uses every observation, even a single extreme value moves it — which is exactly why the median, which ignores extremes, is the better summary for such data.

Reference: NCERT Class 11 Statistics for Economics textbook, chapter Measures of Central Tendency.

Official source: download the NCERT textbook free from ncert.nic.in.