Measures of Central Tendency Class 11

Q: Q10 Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, x, 10, 5, and the mean is 20. Find x and the median.

The missing frequency is 10 and the median is 20. Mean: (450 + 15x) ÷ (20 + x) = 20 gives x = 10. Median: N = 30, so N ÷ 2 = 15. Cumulative frequencies are 5, 15, 25, 30; the 15th item is at the end of class 10–20. Median = 10 + (15 − 5) ÷ 10 × 10 = 20. Final answer: x = 10; median = 20.

Key takeaways

Mean = ΣX ÷ N; the deviations from it sum to zero and it is hit by extreme values.
Median is the middle value; continuous formula L + (N/2 − c.f.)/f × h.
Quartiles split data into four parts, percentiles into hundred.
Mode is the most frequent value; continuous formula L + D1/(D1 + D2) × h.
Choose the average by the data: mean for symmetric, median for skewed, mode for the commonest.

This is Chapter 5 of the Statistics for Economics chapter list for Class 11 Economics. It summarises the tables and graphs of Chapters 3 and 4.

5.1Arithmetic Mean

The arithmetic mean is the sum of all observations divided by their number. It uses every value, is easy to calculate and is the most common average. Its drawbacks are that extreme values pull it up or down and it cannot be found for open-ended classes without an assumption.

Ungrouped data

A stall sells 42, 55, 48, 60, 51, 47, 57 and 40 units on eight days. The direct method gives mean = 400 ÷ 8 = 50. In the assumed mean method we pick A = 50, find deviations (−8, 5, −2, 10, 1, −3, 7, −10), which sum to 0, and so mean = A + Σd ÷ N = 50. The method saves effort when values are large.

Arithmetic mean

Direct: X̄ = ΣX ÷ N · Assumed mean: X̄ = A + Σd ÷ N · Step deviation: X̄ = A + (Σfd′ ÷ N) × h

Grouped data

For a frequency distribution use class marks m: X̄ = Σfm ÷ Σf. For the study-hours data of Chapter 3, class marks 2, 6, 10, 14, 18 and frequencies 1, 9, 11, 8, 1 give Σfm = 296 and mean = 296 ÷ 30 = 9.87 hours. The step deviation method gives the same result: with A = 10 and h = 4, Σfd′ = −1, so mean = 10 + (−1 ÷ 30) × 4 = 9.87.

Properties

The sum of deviations from the mean is zero: Σ(X − X̄) = 0.
It is affected by extreme values.
It depends on every observation.
Changing every value by the same amount changes the mean by that amount.

Weighted arithmetic mean

When items differ in importance we weight them: X̄_w = ΣwX ÷ Σw. A student scoring 70, 60 and 80 in three papers weighted 5, 3 and 2 has a weighted mean of (350 + 180 + 160) ÷ 10 = 69, whereas the simple mean is 70; the weights pull the result towards the heavier paper. Price index numbers in Chapter 7 are weighted averages.

Figure 5.1: The mean reacts to an extreme value; the median does notAdding one festival day of 200 units lifts the mean from 50 to 67.5, but the median stays at 49.5.

5.2Median

The median is the middle value of data arranged in order, so half the observations lie below it and half above. It is not affected by extreme values and can be found for open-ended classes. For an ordered series of N items its position is the (N + 1) divided by 2 th item.

Individual and discrete series

Arrange the eight sales values: 40, 42, 47, 48, 51, 55, 57, 60. With N = 8 the position is (8 + 1) ÷ 2 = 4.5, so the median is the average of the 4th and 5th values: (48 + 51) ÷ 2 = 49.5. For a discrete series use cumulative frequency. If 30 families have 0, 1, 2, 3, 4 children with frequencies 4, 9, 10, 5, 2, the cumulative frequencies are 4, 13, 23, 28, 30. The position (30 + 1) ÷ 2 = 15.5 first appears in 23, so the median is 2 children.

Continuous series

Median of a continuous series

Median = L + [(N ÷ 2 − c.f.) ÷ f] × h

L = lower limit of the median class; c.f. = cumulative frequency before it; f = its frequency; h = its width. The median class holds the (N ÷ 2)th item.

For the study-hours data N ÷ 2 = 15, so the median class is 8–12. Then median = 8 + (15 − 10) ÷ 11 × 4 = 9.82 hours: half the students study fewer hours than this.

5.3Quartiles and Percentiles

Quartiles divide ordered data into four equal parts: the first quartile Q1 has 25 per cent of items below it, Q2 is the median and Q3 has 75 per cent below it. Percentiles divide the data into 100 parts, so the 90th percentile has 90 per cent of items below it.

Positions in an ordered series: Q1 at the (N + 1) ÷ 4 th item and Q3 at 3(N + 1) ÷ 4 th item. For the eight sales values Q1 sits at the 2.25th item: 42 + 0.25 × (47 − 42) = 43.25. Q3 sits at the 6.75th item: 55 + 0.75 × (57 − 55) = 56.5. So the middle half of days had sales between 43.25 and 56.5.

The k-th percentile is at k(N + 1) ÷ 100 th position. A candidate at the 82nd percentile in a test did better than about 82 per cent of those who sat it, and was below about 18 per cent.

Quartiles and percentiles are called positional averages because they depend on position in the ordered data, not on all values.

5.4Mode

The mode is the value that occurs most often in a data set. It is the most typical value, useful for qualitative data such as the most popular shoe size. A distribution may have one mode, two modes or none. For grouped data, the modal class is the class with the highest frequency.

In 1, 2, 3, 4, 4, 5 the mode is 4. In 1, 1, 2, 2, 3, 3 no value stands out, and in 2, 2, 5, 5, 7 there are two modes (bimodal). In the children example the largest frequency is 10, so the mode is 2 children.

Mode of a continuous series

Mode = L + [D₁ ÷ (D₁ + D₂)] × h

L = lower limit of the modal class; D₁ = its frequency minus the frequency of the class before it; D₂ = its frequency minus that of the class after it; h = class width.

For the study-hours data the modal class is 8–12 (frequency 11). D₁ = 11 − 9 = 2, D₂ = 11 − 8 = 3, so mode = 8 + 2 ÷ 5 × 4 = 9.60 hours. The formula needs equal class widths and exclusive classes.

Bimodal data

A data set with two peaks is called bimodal. If a shop records the ages of its customers and finds peaks around 18 and around 45, the mode alone is misleading, and the mean, around 32, describes nobody. Such a pattern usually means that two different groups have been mixed, and it is better to study the groups separately.

This is a reminder that an average is only a summary. It should always be read together with a table or graph showing how the values are spread.

5.5Relative Position of Mean, Median and Mode and Choosing an Average

In a moderately skewed distribution the median lies between the mean and the mode, so the three appear in alphabetical order or its reverse. An empirical relation is mode = 3 median minus 2 mean. Choose the mean for symmetric data, the median when extremes distort, and the mode for the most common category.

Figure 5.2: Mean, median and mode of one distributionFor the study-hours data: mode 9.60, median 9.82, mean 9.87. The median lies between the other two; 3 × 9.82 − 2 × 9.87 = 9.72, close to the calculated mode.

Which average to use
Situation	Suitable average	Reason
Symmetric numerical data	Arithmetic mean	Uses every value
Data with a few very large or small values	Median	Not distorted by extremes
Open-ended class at one end	Median	Mean needs a class mark
Most popular size, brand or colour	Mode	Shows the commonest choice
Items of unequal importance	Weighted mean	Gives each its weight

Skewness and the three averages

A distribution is symmetric if its left and right halves mirror each other. In that case the mean, median and mode are equal. If a few very large values stretch the right tail, as with incomes, the distribution is positively skewed and the mean is greater than the median, which is greater than the mode. If a few very small values stretch the left tail, the reverse holds.

This is why average income is higher than the income of the typical household, and why news reports often quote the median. For the study-hours data, the three averages are close together (9.60, 9.82 and 9.87), which tells us that the distribution is nearly symmetric.

Worked Examples

Example 1: Mean by the assumed mean method

Given: Daily sales: 42, 55, 48, 60, 51, 47, 57, 40.
Formula: X̄ = A + Σd ÷ N with A = 50
Steps: d = −8, 5, −2, 10, 1, −3, 7, −10; Σd = 0. Mean = 50 + 0 ÷ 8.
Answer: The mean is 50 units a day.

Example 2: Mean, median and mode of grouped data

Given: Classes 0–4, 4–8, 8–12, 12–16, 16–20 with frequencies 1, 9, 11, 8, 1.
Formula: Mean = Σfm ÷ N; median and mode by the class formulas
Steps: Σfm = 2 + 54 + 110 + 112 + 18 = 296, mean = 296 ÷ 30 = 9.87. Median = 8 + (15 − 10) ÷ 11 × 4 = 9.82. Mode = 8 + 2 ÷ 5 × 4 = 9.60.
Answer: Mean 9.87, median 9.82, mode 9.60 hours.

Example 3: Missing frequency

Given: Classes 0–10, 10–20, 20–30, 30–40 with frequencies 5, x, 10, 5; mean = 20.
Formula: Mean = Σfm ÷ Σf with marks 5, 15, 25, 35
Steps: Σfm = 25 + 15x + 250 + 175 = 450 + 15x; Σf = 20 + x. So 450 + 15x = 20(20 + x) = 400 + 20x, giving 5x = 50.
Answer: x = 10. Check: N = 30, Σfm = 600, mean = 20.

Example 4: Quartiles

Given: Ordered sales: 40, 42, 47, 48, 51, 55, 57, 60.
Formula: Q1 at (N + 1) ÷ 4; Q3 at 3(N + 1) ÷ 4
Steps: Q1 at 2.25th item = 42 + 0.25 × 5 = 43.25. Q3 at 6.75th item = 55 + 0.75 × 2 = 56.5.
Answer: Q1 = 43.25 and Q3 = 56.5.

Common Mistakes

Using (N + 1) ÷ 2 for the median class of a continuous series. Use N ÷ 2 there.
Forgetting to arrange data in order before finding the median or quartiles.
Taking the simple mean of ratios or rates when weights differ.
Applying the mode formula with unequal class widths or inclusive classes.

10 Important Questions for Class 11 Board and School Examinations

Questions are written in original words. Marks are shown beside each question.

Q1The average most affected by extreme values is (a) median (b) mode (c) arithmetic mean (d) quartile 1 mark · MCQ

The correct option is (c) arithmetic mean.

Q2The sum of deviations of items from their arithmetic mean is (a) positive (b) negative (c) zero (d) one 1 mark · MCQ

The correct option is (c) zero.

Q3Assertion (A): The median is a better average for income data. Reason (R): It is not affected by a few very high incomes. Which is correct? 1 mark · Assertion-reason

Both A and R are true, and R explains A.

Q4State any three properties of the arithmetic mean. 3 marks

The arithmetic mean uses all values, is affected by extreme values and balances deviations to zero.

The sum of deviations of items from the mean is zero.
It depends on every observation in the data.
Extreme values pull it up or down.
If every value changes by a constant, the mean changes by that constant.

These properties explain both its usefulness and its main weakness.

Q5Which average would you use for the most popular shoe size? Why? 3 marks

The mode is the suitable average for the most popular shoe size, because it shows the size sold most often.

The mean size, say 7.4, may not correspond to a size anyone wears, and the median may also fall between sizes. The mode is a real size with the highest sales, so it tells the shop which size to stock most. The mode is also the only average that suits qualitative choices such as brand or colour.

Q6Find the median of 12, 7, 15, 9, 20, 11, 14, 6. 4 marks

The median of 12, 7, 15, 9, 20, 11, 14 and 6 is 11.5.

Arrange in order: 6, 7, 9, 11, 12, 14, 15, 20.
N = 8, so the median is the average of the (N ÷ 2)th and the next item.
The 4th and 5th values are 11 and 12.
Median = (11 + 12) ÷ 2 = 11.5.

Half of the values lie below 11.5 and half above.

The method uses the position (N + 1) ÷ 2 = 4.5, which falls between the 4th and 5th items.

Q7A student scores 70, 60 and 80 in three papers carrying weights 5, 3 and 2. Find the weighted mean. 4 marks

The weighted mean score is 69.

Given: scores 70, 60 and 80 with weights 5, 3 and 2.
Formula: weighted mean = ΣwX ÷ Σw.
Steps: (70 × 5 + 60 × 3 + 80 × 2) ÷ (5 + 3 + 2) = (350 + 180 + 160) ÷ 10 = 690 ÷ 10.
Answer: 69, which is lower than the simple mean of 70 because the paper with the largest weight has the lowest score among the two heavier ones.

Weights pull the average towards the more important items.

Q8Explain how the median of a continuous series is found. Illustrate with a short example. 6 marks

To find the median of a continuous series, form cumulative frequencies, locate the median class and apply the median formula.

Median class: the class containing the (N ÷ 2)th item.
Formula: Median = L + [(N ÷ 2 − c.f.) ÷ f] × h.
L is the lower limit of the median class, c.f. the cumulative frequency before it, f its frequency and h its width.
Example: for N = 30 with frequencies 1, 9, 11, 8 and 1 over classes of width 4 starting at 0, N ÷ 2 = 15 falls in the class 8–12.
Median = 8 + (15 − 10) ÷ 11 × 4 = 9.82.

Half the items therefore lie below 9.82.

The same formula works for unequal class widths.

Q9Compare mean, median and mode: meaning, merits and demerits. 6 marks

The mean is the arithmetic average, the median the middle item and the mode the most frequent item, and each has merits and demerits.

Mean: uses every value and is easy to calculate, but is distorted by extreme values.
Median: not affected by extremes and usable with open-ended classes, but it ignores most values and needs ordered data.
Mode: simple, shows the most typical value and suits categories, but may be absent or not unique.
In a moderately skewed distribution the median lies between the mean and the mode.
Choose by the purpose and shape of the data.

For incomes, the median is usually the most informative single figure.

The three are connected: for a symmetric distribution they coincide, and in a skewed one the mean is pulled towards the long tail, so the median is a safer summary for incomes.

Q10Classes 0–10, 10–20, 20–30, 30–40 have frequencies 5, x, 10, 5, and the mean is 20. Find x and the median. 4 marks · Numerical

The missing frequency is 10 and the median is 20.

Mean: (450 + 15x) ÷ (20 + x) = 20 gives x = 10.

Median: N = 30, so N ÷ 2 = 15. Cumulative frequencies are 5, 15, 25, 30; the 15th item is at the end of class 10–20. Median = 10 + (15 − 5) ÷ 10 × 10 = 20.

Final answer: x = 10; median = 20.

10 Points for State PSC and UPSC Aspirants

Per capita income is an arithmetic mean and does not show how income is distributed.Why it matters: the standard critique of averages in development economics.Source: NCERT; Economic Survey
In right-skewed income data the mean exceeds the median, so the median describes a typical household better.Why it matters: applies to household income and wealth data.Source: NCERT
The Periodic Labour Force Survey reports average earnings of workers, a mean with its own limitations.Why it matters: check what kind of average a report uses.Source: MoSPI
NSS consumption reports present spending by fractile classes, positional groups like quartiles.Why it matters: ties quartiles to inequality reports.Source: MoSPI
Weighted averages underlie the Consumer Price Index and Wholesale Price Index.Why it matters: links this chapter to Chapter 7.Source: MoSPI; Office of the Economic Adviser
The geometric mean is the correct average for growth rates; compound annual growth rate is a geometric mean.Why it matters: growth questions in Economic Survey use it.Source: NCERT; standard economics textbooks
Core inflation removes volatile items such as food and fuel, much as the median ignores extreme values.Why it matters: a policy analogue of a robust average.Source: Reserve Bank of India
Entrance examinations report percentile scores, which show relative position, not marks.Why it matters: a daily use of percentiles.Source: NCERT
The Gini coefficient and Lorenz curve measure how unequal a distribution is around its average.Why it matters: an averages-and-inequality question.Source: NCERT; World Bank
Mean > median > mode signals positive skewness; the reverse signals negative skewness.Why it matters: frequently asked as a single-line fact.Source: NCERT

These are revision notes, not official predictions of exam content.

Quick Revision

Mean = ΣX ÷ N; Σ(X − X̄) = 0.
Median position: (N + 1) ÷ 2; continuous: N ÷ 2.
Median = L + (N/2 − c.f.)/f × h.
Mode = L + D1/(D1 + D2) × h.
Q1 at (N + 1)/4; Q3 at 3(N + 1)/4.
Mean is hit by extremes; median is not.
Mode = 3 median − 2 mean (approximate).

Glossary:

Arithmetic mean Median Mode Quartile Weighted mean

Related chapters: Presentation of Data, which draws the same distribution; Correlation, which studies two variables at once.

Sources

NCERT textbooks: Statistics for Economics, Class 11, National Council of Educational Research and Training. Syllabus reference. Accessed 23 October 2026.
Ministry of Statistics and Programme Implementation (MoSPI), Government of India. Accessed 23 October 2026.
Economic Survey, Ministry of Finance, Government of India. Accessed 23 October 2026.
Reserve Bank of India. Accessed 23 October 2026.
Press Information Bureau, Government of India. Accessed 23 October 2026.

About this page

Drafted with AI assistance from the NCERT chapter structure, in original words. Content is checked and approved by Dr. Anamika Shrivastava, PhD Economics, before publication. Read how this site works or send a correction.

Syllabus year 2026-27 · Last updated 23 October 2026