CBSE Class 11 Statistics Chapter 5: Measures of Central Tendency NCERT Solutions
CBSE Class 11 Statistics introduces the crucial concept of Measures of Central Tendency. This chapter is all about understanding how to summarize large sets of data using a single representative value, often called an average. We explore different types of averages, focusing on the Arithmetic Mean, Median, and Mode. You'll learn not just how to calculate these, but also when each one is the most suitable measure depending on the nature of your data. For instance, we'll discuss how to handle qualitative data, data with outliers, and distributions that don't have a clear upper or lower limit. The chapter also covers important properties of these averages, such as the sum of deviations from the mean always being zero. These insights are key to interpreting data effectively and will be invaluable for your exams.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 11 |
| Subject | Statistics (Economics) |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | 5. Measures of Central Tendency |
Chapter summary
Chapter 5 of the CBSE Class 11 Statistics textbook focuses on Measures of Central Tendency. This section of NCERT Solutions provides clear explanations and answers for exercises related to identifying the most suitable average (Mean, Median, or Mode) for various scenarios, including qualitative data, open-ended distributions, and data with extreme values. It also covers key properties of the Arithmetic Mean, such as the sum of deviations. These solutions aim to build a strong foundation in understanding and applying central tendency measures.
Learning outcomes
- Understand the concept and purpose of measures of central tendency.
- Identify the most suitable average (Mean, Median, Mode) for different types of data.
- Explain the properties of Arithmetic Mean, Median, and Mode.
- Solve problems involving the calculation and application of central tendency measures.
- Differentiate between qualitative and quantitative data in the context of averages.
Topics covered
Paper topics
- Measures of Central Tendency
- Arithmetic Mean
- Median
- Mode
- Suitability of Averages
- Qualitative Measurements
- Quantitative Measurements
- Extreme Values
- Open-ended Frequency Distributions
- Sum of Deviations from Mean
Important topics
- Identifying the most suitable average for different data types
- Properties of Arithmetic Mean
- Application of Median in open-ended distributions
- Impact of extreme values on Mean vs. Median/Mode
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Questions and Solutions
Question 1
Which average would be suitable in the following cases?
(i) Average size of readymade garments.
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wage in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii) In case of open-ended frequency distribution.
- (i) Mode: For average size of readymade garments, the Mode is the most suitable average. This is because it represents the most frequently occurring size, which is most relevant for inventory and sales purposes.
- (ii) Median: For average intelligence of students in a class, the Median is preferred. Intelligence is a qualitative characteristic, and the Median represents the middle value, dividing the students into two equal halves, making it suitable for such non-numeric or ordinal data.
- (iii) Arithmetic Mean: For average production in a factory per shift, the Arithmetic Mean is suitable. Production is a quantitative measure, and the Mean provides a precise average of the output over different shifts.
- (iv) Arithmetic Mean: For average wage in an industrial concern, the Arithmetic Mean is appropriate. Wages are quantitative, and the Mean gives a clear indication of the typical earnings per worker.
- (v) Arithmetic Mean: When the sum of absolute deviations from an average is least, that average is the Arithmetic Mean. This is a defining property of the Mean, where \(\sum_{i=1}^{n} |x_i - \bar{x}|\) is minimized.
- (vi) Arithmetic Mean: When quantities of the variable are expressed in ratios, the Arithmetic Mean is suitable. Ratios represent quantitative relationships, and the Mean can effectively summarize these values.
- (vii) Median: In the case of an open-ended frequency distribution (where class intervals at the beginning or end are not defined), the Median is generally used. This is because the Median calculation relies on the position of the middle value and does not require the exact class limits or magnitudes for open-ended classes.
Question 2
Indicate the most appropriate alternative from the multiple choices provided against each question.
(i) The most suitable average for qualitative measurement is
(a) Arithmetic mean
(b) Median
(c) Mode
(d) Geometric mean
(e) None of the above
(ii) Which average is affected most by the presence of extreme items?
(a) Median
(b) Mode
(c) Arithmetic mean
(d) None of the above
(iii) The algebraic sum of deviation of a set of n values from A.M is
(a) N
(b) 0
(c) 1
(d) none of the above
- (i) The most suitable average for qualitative measurement is (b) Median. While Mode can sometimes be used for qualitative data (like favorite color), the Median is often preferred when trying to establish a central point in ordered qualitative categories or when dealing with data that is not strictly numerical but can be ranked.
- (ii) The average most affected by the presence of extreme items is (c) Arithmetic mean. Extreme values (very high or very low) can significantly pull the Arithmetic Mean towards them, thus distorting the representation of the typical value in the dataset. The Median and Mode are less affected by such extreme values.
- (iii) The algebraic sum of deviations of a set of n values from A.M is (b) 0. This is a fundamental property of the Arithmetic Mean. For any dataset \(x_1, x_2, ..., x_n\) with mean \(\bar{x}\), the sum \(\sum_{i=1}^{n} (x_i - \bar{x}) = 0\).
Common mistakes
- Choosing the wrong measure of central tendency for qualitative data.
- Not recognizing the impact of extreme values on the Arithmetic Mean.
- Confusing the properties of Mean, Median, and Mode.
- Incorrectly applying Median to open-ended frequency distributions.
Revision tips
- Focus on understanding the conditions under which each measure (Mean, Median, Mode) is most appropriate.
- Memorize the key properties of each average, especially the sum of deviations from the Mean.
- Practice identifying the type of data (qualitative, quantitative, open-ended) to select the correct average.
- Review the explanations for each question to reinforce the reasoning behind the chosen average.
Practice MCQs
Q1. Which measure of central tendency is most suitable for qualitative measurements like 'average intelligence'?
Explanation: Median is preferred for qualitative data as it represents the middle value and is less affected by the nature of the categories compared to the mean.
Q2. The presence of extreme values in a dataset most significantly affects which average?
Explanation: The Arithmetic Mean is calculated by summing all values, so extreme high or low values can disproportionately pull the mean towards them.
Q3. What is the algebraic sum of deviations of a set of n values from their Arithmetic Mean?
Explanation: A fundamental property of the Arithmetic Mean is that the sum of the differences between each data point and the mean is always zero.
Q4. For determining the average size of readymade garments, which measure is most suitable?
Explanation: Mode is the most frequent value, making it ideal for identifying the most common garment size sold in the market.
Q5. In an open-ended frequency distribution, which average is generally preferred?
Explanation: Median can be calculated for open-ended distributions without needing to know the exact class sizes or magnitudes, unlike the Arithmetic Mean.
Frequently asked questions
What are the main measures of central tendency covered in this chapter?
This chapter covers the Arithmetic Mean, Median, and Mode as the primary measures of central tendency, discussing their properties and applications.
When is the Median the most suitable average to use?
The Median is most suitable for qualitative data, in open-ended frequency distributions, and when the sum of absolute deviations from the median is least.
Why is the Arithmetic Mean affected most by extreme items?
The Arithmetic Mean is calculated by summing all values, so very large or very small (extreme) values can significantly pull the mean towards them.
What is a key property of the Arithmetic Mean regarding deviations?
The algebraic sum of deviations of all values in a dataset from their Arithmetic Mean is always zero.
How do these NCERT Solutions help in exam preparation?
These solutions provide clear, rewritten explanations for each question, helping students understand the concepts of central tendency and how to apply them, which is crucial for exam success.
Which average is best for finding the most common value in a dataset?
The Mode is the best average for finding the most common or frequently occurring value in a dataset, such as the most popular shoe size.
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