Measures of Central Tendency Class 10 ICSE — Bansal Ch 24
Measures of central tendency class 10 ICSE maths starts from one question: which single value best represents a whole group? Chapter 24 of the Bansal ICSE Mathematics book answers it with three averages — arithmetic mean, median and mode — then adds quartiles and the inter-quartile range. Each section builds on the last, from raw data to frequency tables to ogives.
Download Measures of Central Tendency Class 10 ICSE Mathematics Chapter 24 PDF
Download the Measures of Central Tendency Class 10 ICSE Mathematics chapter PDF from the Bansal textbook to revise offline or print for practice. It is the full chapter as printed in the book — 29 pages, 11 sections, 76 questions.
What is inside Chapter 24
29 pages · 11 sections · 13 worked examples · 76 exercise questions
| Section | Title |
|---|---|
| 24.1 | Introduction |
| 24.2 | Arithmetic Mean |
| 24.3 | Arithmetic Mean of Tabulated Data |
| 24.4 | To Find Mean for Grouped Data (both continuous and discontinuous) |
| 24.5 | Median |
| 24.6 | Median for Raw Data |
| 24.7 | Median for Tabulated Data |
| 24.8 | Median for Grouped Data (both continuous and discontinuous) |
| 24.9 | Quartiles |
| 24.10 | Inter-quartile Range |
| 24.11 | Mode |
- EXERCISE 24(A): 17 questions
- EXERCISE 24(B): 11 questions
- EXERCISE 24(D): 48 questions
24.1 Introduction
Measures of central tendency are numerical expressions that represent the characteristics of a group. A group may hold hundreds of observations. Its average condenses all of them into one number.
A good average is not the lowest or the highest value in the group. It sits somewhere between the two limits, near the centre, where most items cluster. That position is why these values are called measures of central tendency.
This ICSE chapter studies three averages:
- Arithmetic mean
- Median
- Mode
Each average answers a different question about the data, so the chapter treats them separately before combining them in mixed problems. For the other chapters in the same book, see the Class 10 ICSE Maths book list.
24.2 Arithmetic Mean
The arithmetic mean of a set of numbers is obtained by dividing the sum of the numbers by the number of numbers. For n numbers x_1, x_2, x_3, \dots, x_n:
\text{Mean} = \frac{x_1 + x_2 + x_3 + \dots + x_n}{n} = \frac{\sum x}{n}
The Greek letter \Sigma (sigma) means “sum of the numbers”. So \sum x is the total of all values in the set.
Worked pattern: six students weigh 48, 52, 55, 49, 51 and 45 kg. Here n = 6 and \sum x = 300 kg. The mean is 300 \div 6 = 50 kg.
The mean shares the total equally among all items. That is why it acts as the balance point of the group.
24.3 Arithmetic Mean of Tabulated Data
For a discrete frequency distribution, the mean can be found by any of three methods: direct method, short-cut method and step-deviation method. All three give the same answer.
Direct method: make columns for the variate x, the frequency f, and the product fx. Add the frequencies to get \Sigma f, add the products to get \Sigma fx, then use
\text{Mean} = \bar{x} = \frac{\sum fx}{\sum f}
The symbol \bar{x} is the usual way to write the mean.
Short-cut method: take an assumed mean A, preferably one of the x values. Form the deviation d = x - A, multiply each deviation by its frequency, and use
\text{Mean} = A + \frac{\sum fd}{\sum f}
Step-deviation method: divide every deviation by a common factor i to get t = \frac{x - A}{i}, then use
\text{Mean} = A + \frac{\sum ft}{\sum f} \times i
Worked pattern (short-cut method), with assumed mean A = 24:
| x | f | d = x - A | fd |
|---|---|---|---|
| 20 | 3 | -4 | -12 |
| 22 | 5 | -2 | -10 |
| A = 24 | 7 | 0 | 0 |
| 26 | 4 | 2 | 8 |
| 28 | 1 | 4 | 4 |
| \Sigma f = 20 | \Sigma fd = -10 |
So \text{Mean} = 24 + \frac{-10}{20} = 24 - 0.5 = 23.5.
Missing frequency: when the mean is given and one frequency f is unknown, write \Sigma f and \Sigma fx in terms of f, then solve \frac{\sum fx}{\sum f} = \text{mean}. Cross-multiply and isolate f.
Pattern: variates 3, 5, 7, 9 with frequencies 6, f, 10, 4 have mean 6. Then \frac{124 + 5f}{20 + f} = 6, so 120 + 6f = 124 + 5f and f = 4.
Edge case: if a frequency contains an unknown such as 15p, the algebra may return two values, for example p = \pm 1. Reject the negative one because a frequency cannot be negative.
Common mistakes in this section:
- Median order: always arrange data in ascending or descending order before picking the middle term.
- Class mark: the class mark is \frac{\text{lower limit} + \text{upper limit}}{2}, not half the difference.
- Missing frequency algebra: multiply both sides of \frac{\sum fx}{\sum f} = \text{mean} by \Sigma f before simplifying.
24.4 To Find Mean for Grouped Data (both continuous and discontinuous)
For grouped data, each class interval is first replaced by its class mark, also called the mid value. The class mark of an interval is
\text{Class mark} = \frac{\text{lower limit} + \text{upper limit}}{2}
For example, the class 10 - 20 has class mark \frac{10 + 20}{2} = 15. The section applies to both continuous and discontinuous intervals; the class mark is the starting point in each case.
Use the class mark as the variate x, then apply the direct, short-cut or step-deviation method exactly as in section 24.3. Whichever method you choose, the mean comes out the same. That matching answer is a useful check on your arithmetic.
Worked pattern: classes 0-10, 10-20, 20-30, 30-40 have frequencies 4, 6, 10, 5. Their class marks are 5, 15, 25, 35 and the products fx are 20, 90, 250, 175. Total \Sigma fx = 535, \Sigma f = 25, so the mean is \frac{535}{25} = 21.4.
Two missing frequencies: when the total number of observations n and the mean are both given, write \Sigma f = n and \frac{\sum fx}{n} = \text{mean}. Each statement gives one equation, and the two equations are solved together for f_1 and f_2.
| Method | Formula | Fastest when |
|---|---|---|
| Direct | \frac{\sum fx}{\sum f} | values are small and few |
| Short cut | A + \frac{\sum fd}{\sum f} | values are large but close together |
| Step deviation | A + \frac{\sum ft}{\sum f} \times i | values are large with equal spacing |
24.5 Median
The median is the value of the middle term of a set of variables, when the variables are arranged in ascending or descending order.
Arrangement is the key step. Until the data is ordered, there is no middle term to identify.
The next three sections show how to apply this definition to raw data, to a frequency table, and to grouped data.
24.6 Median for Raw Data
For raw data with n terms arranged in ascending or descending order:
- n odd: median is the \left(\frac{n+1}{2}\right)^{\text{th}} term.
- n even: median is the mean of the \left(\frac{n}{2}\right)^{\text{th}} and \left(\frac{n}{2}+1\right)^{\text{th}} terms.
Worked pattern (odd): the marks 12, 7, 15, 9, 11, arranged ascending, are 7, 9, 11, 12, 15. With n = 5, the median is the 3rd term, 11.
Worked pattern (even): the heights 10, 4, 8, 6, 14, 2, 12, 16, arranged ascending, are 2, 4, 6, 8, 10, 12, 14, 16. With n = 8, the median is \frac{8 + 10}{2} = 9.
Note: ascending or descending order gives the same median. A common mistake is reading the middle value before ordering the data; the fix is to arrange first.
If the median is given and an unknown x is hidden among the terms, write the median formula in terms of x, then solve the equation.
24.7 Median for Tabulated Data
For a discrete frequency table, add a cumulative frequency column. Each entry is the running total of frequencies up to that variate.
For n odd, the median is the \left(\frac{n+1}{2}\right)^{\text{th}} term. Look down the cumulative column until the running total reaches or passes this position. The variate on that row is the median.
Worked pattern: variates 40, 42, 44, 46 have frequencies 4, 8, 10, 9. Cumulative frequencies are 4, 12, 22, 31. Here n = 31, so the median is the 16th term. The cumulative column shows terms 13 to 22 all fall in the variate 44. The median is 44.
24.8 Median for Grouped Data (both continuous and discontinuous)
For grouped data, the median is read from an ogive, a free-hand cumulative frequency curve.
Steps:
- Build the cumulative frequency table.
- Plot points of upper class limit against cumulative frequency.
- Draw a smooth free-hand curve through the points.
To read the median, mark the \left(\frac{n}{2}\right)^{\text{th}} or \left(\frac{n+1}{2}\right)^{\text{th}} term on the y-axis. Draw a horizontal line to the curve, then a vertical line down to the x-axis. The value at that point is the median.
Worked pattern: 80 workers have wages grouped in classes of 10. With n = 80, the median is the 40th term. If the ogive meets the x-axis at 42, the median wage is about \text{₹}42.
The class interval containing the median is called the median class. Board questions often ask for more: to find how many workers earn more than a set wage, read the cumulative frequency up to that wage, subtract it from n, and convert to a percentage. The exam pattern draws these questions from the official CISCE website syllabus.
24.9 Quartiles
Quartiles are the values of the variate that divide the ordered data into four equal parts. The middle quartile Q_2 is the same as the median. Q_1 is the lower quartile and Q_3 the upper quartile.
For n terms arranged ascending:
- n odd: Q_1 = \left(\frac{n+1}{4}\right)^{\text{th}} term, Q_3 = \left(\frac{3(n+1)}{4}\right)^{\text{th}} term.
- n even: Q_1 = \left(\frac{n}{4}\right)^{\text{th}} term, Q_3 = \left(\frac{3n}{4}\right)^{\text{th}} term.
Mnemonic: the odd one carries the extra one. For odd n, add 1 before dividing by 4. For even n, use n as it is.
Worked pattern: the data 3, 8, 10, 12, 15, 18, 20 has n = 7. Here Q_1 is the 2nd term, 8, and Q_3 is the 6th term, 18.
24.10 Inter-quartile Range
The inter-quartile range is the difference between the upper and lower quartiles:
\text{Inter-quartile range} = Q_3 - Q_1
It is always positive, because Q_3 is always greater than Q_1. For the data 3, 8, 10, 12, 15, 18, 20, the range is 18 - 8 = 10.
The quartiles can also be read from an ogive. Locate the \left(\frac{n}{4}\right)^{\text{th}} term on the y-axis for Q_1, and the \left(\frac{3n}{4}\right)^{\text{th}} term for Q_3. Read both x-values from the curve, then subtract.
Worked pattern: for a distribution of 120 scores with n = 120, read Q_1 at the 30th term and Q_3 at the 90th term. If the ogive gives 28 and 61, the inter-quartile range is 61 - 28 = 33.
24.11 Mode
The mode is the value that occurs most frequently in a set of observations. It is the point of maximum frequency.
For raw data, count how many times each value appears. The value with the highest count is the mode. Worked pattern: in 5, 6, 5, 7, 5, 8, 6, the value 5 appears three times, so the mode is 5.
For a discrete frequency table, look at the frequency column. The variate with the highest frequency is the mode.
For grouped data, draw a histogram. The highest rectangle is the modal class. Join the upper corners of the rectangles next to it with two diagonals. From their intersection, drop a perpendicular to the x-axis. The point it meets is the mode.
Worked pattern: if the tallest rectangle is the class 50-60, the modal class is 50-60, and the perpendicular falls at about 53, the mode.
What to practise from Measures of Central Tendency Class 10 ICSE
This ICSE chapter has three exercise blocks. Work them in the order given below, because each one builds on the skills of the earlier ones.
- Exercise 24(A) — 17 questions on the mean of raw and tabulated data, including the three methods and missing frequencies.
- Exercise 24(B) — 11 questions on the mean of grouped data, using direct, short-cut and step-deviation methods.
- Exercise 24(D) — 48 questions mixing mean, median, quartiles, mode and ogives, including board questions.
There is no Exercise 24(C) in this chapter, so do not look for one. For the full set of subjects, see all ICSE Class 10 books.
Exam pattern: board questions from 2002 to 2015 are scattered through these blocks, marked with the year in brackets. Ogive questions asking for the median, quartiles or inter-quartile range appear almost every year, and mean questions recur from 2005 onwards. Practising the year-labelled questions first shows you the repeated patterns.
Measures of Central Tendency Class 10 ICSE frequently asked questions
What are measures of central tendency class 10 ICSE?
They are numerical expressions that represent the characteristics of a whole group of data. Chapter 24 studies three of them — the arithmetic mean, the median and the mode — and then adds quartiles and the inter-quartile range.
In Class 10 ICSE Maths, how do I find the median when the number of terms is even?
Arrange the data in ascending or descending order first. Then take the mean of the \left(\frac{n}{2}\right)^{\text{th}} and \left(\frac{n}{2}+1\right)^{\text{th}} terms.
What is the difference between the short cut method and the step deviation method in this ICSE chapter?
The short cut method uses deviations d = x - A from an assumed mean. The step deviation method divides those deviations by a common factor i to get t = \frac{x - A}{i}, then multiplies the result back by i at the end.
How do I read the median and quartiles from an ogive in measures of central tendency?
Plot upper class limits against cumulative frequency and draw a free-hand curve. Use the \left(\frac{n}{2}\right)^{\text{th}} term for the median, the \left(\frac{n}{4}\right)^{\text{th}} term for Q_1, and the \left(\frac{3n}{4}\right)^{\text{th}} term for Q_3 on the y-axis, then read the matching x-values.
How do I find a missing frequency when the mean is given in the Class 10 ICSE syllabus?
Write both \Sigma f and \Sigma fx in terms of the unknown frequency. Set \frac{\sum fx}{\sum f} equal to the given mean, cross-multiply and solve. If the solution makes a frequency negative, reject it.
Reference: ICSE Class 10 Mathematics textbook material, chapter 24: Measures of Central Tendency — CISCE-aligned.
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