CBSE Class 11 Mathematics Chapter 15: Statistics NCERT Solutions

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NCERT Solutions PDF Class 11 PDF

This chapter delves into the fundamental concepts of Statistics for Class 11 Mathematics, focusing on measures of dispersion. The NCERT Solutions for Chapter 15 provide detailed, step-by-step explanations for calculating the Mean Deviation about the Mean and the Mean Deviation about the Median. These solutions are designed to help students understand the process of organizing raw data, calculating the arithmetic mean and median, finding the absolute deviations from these central tendencies, and finally computing the mean deviation. By working through these examples, students will develop a strong grasp of these statistical measures, crucial for data analysis and interpretation. The solutions are presented clearly, making them an excellent resource for exam preparation and revision, ensuring students can confidently tackle similar problems.

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Quick info

BoardCBSE
ClassClass 11
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 15

Chapter summary

Chapter 15 of the CBSE Class 11 Mathematics syllabus focuses on Statistics, specifically introducing measures of dispersion. This section of NCERT Solutions provides detailed guidance on calculating the Mean Deviation about the Mean and the Mean Deviation about the Median. It covers the essential steps involved, from finding the central tendency (mean or median) to calculating absolute deviations and averaging them. These solutions are key for students to practice and master the computation of these dispersion measures.

Learning outcomes

  • Understand the concept of Mean Deviation.
  • Calculate the Mean Deviation about the Mean for a given dataset.
  • Calculate the Mean Deviation about the Median for a given dataset.
  • Interpret the meaning of Mean Deviation in the context of data spread.
  • Apply the formulas for Mean Deviation accurately.

Topics covered

Paper topics

  • Statistics
  • Measures of Dispersion
  • Mean Deviation
  • Mean Deviation about the Mean
  • Mean Deviation about the Median
  • Arithmetic Mean Calculation
  • Median Calculation
  • Absolute Deviations

Important topics

  • Mean Deviation about the Mean
  • Mean Deviation about the Median
  • Calculation of Mean
  • Calculation of Median
  • Absolute Value of Deviations

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Questions and Solutions

Question 1

Find the mean deviation about the mean for the data:

4, 7, 8, 9, 10, 12, 13, 17

Solution:

The given data set is: 4, 7, 8, 9, 10, 12, 13, 17.

First, we calculate the arithmetic mean (\overline{x}) of the data. There are 8 observations.

\overline{x} = \frac{4+7+8+9+10+12+13+17}{8} = \frac{80}{8} = 10

Next, we find the deviation of each observation from the mean (x_i - \overline{x}):

-6, -3, -2, -1, 0, 2, 3, 7

Now, we find the absolute values of these deviations (|x_i - \overline{x}|):

6, 3, 2, 1, 0, 2, 3, 7

Finally, we calculate the Mean Deviation about the Mean (M.D.) by taking the average of these absolute deviations:

M.D.(\overline{x}) = \frac{\sum_{i=1}^{8} |x_i - \overline{x}|}{8} = \frac{6+3+2+1+0+2+3+7}{8} = \frac{24}{8} = 3

The Mean Deviation about the mean is 3.

Question 2

Find the mean deviation about the mean for the data:

38, 70, 48, 40, 42, 55, 63, 46, 54, 44

Solution:

The given data set is: 38, 70, 48, 40, 42, 55, 63, 46, 54, 44.

There are 10 observations. We first calculate the arithmetic mean (\overline{x}):

\overline{x} = \frac{38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44}{10} = \frac{500}{10} = 50

Next, we find the deviation of each observation from the mean (x_i - \overline{x}):

-12, 20, -2, -10, -8, 5, 13, -4, 4, -6

Now, we find the absolute values of these deviations (|x_i - \overline{x}|):

12, 20, 2, 10, 8, 5, 13, 4, 4, 6

Finally, we calculate the Mean Deviation about the Mean (M.D.) by averaging these absolute deviations:

M.D.(\overline{x}) = \frac{\sum_{i=1}^{10} |x_i - \overline{x}|}{10} = \frac{12+20+2+10+8+5+13+4+4+6}{10} = \frac{84}{10} = 8.4

The Mean Deviation about the mean is 8.4.

Question 3

Find the mean deviation about the median for the data:

13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

Solution:

The given data set is: 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17.

There are 12 observations. To find the median, we first arrange the data in ascending order:

10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18

Since the number of observations (n=12) is even, the median (M) is the average of the \frac{n}{2}th and (\frac{n}{2} + 1)th observations.

Median M = Average of the 6th and 7th observations.

M = \frac{13 + 14}{2} = \frac{27}{2} = 13.5

Now, we find the deviation of each observation from the median (x_i - M):

-3.5, -2.5, -2.5, -1.5, -0.5, -0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

Next, we find the absolute values of these deviations (|x_i - M|):

3.5, 2.5, 2.5, 1.5, 0.5, 0.5, 0.5, 2.5, 2.5, 3.5, 3.5, 4.5

Finally, we calculate the Mean Deviation about the Median (M.D.) by taking the average of these absolute deviations:

M.D.(M) = \frac{\sum_{i=1}^{12} |x_i - M|}{12} = \frac{3.5+2.5+2.5+1.5+0.5+0.5+0.5+2.5+2.5+3.5+3.5+4.5}{12}

M.D.(M) = \frac{28}{12} = \frac{7}{3} \approx 2.33

The Mean Deviation about the median is \frac{7}{3} or approximately 2.33.

Common mistakes

  • Errors in calculating the arithmetic mean or median.
  • Forgetting to take the absolute value of deviations.
  • Incorrectly summing the absolute deviations.
  • Mistakes in ordering the data for median calculation.
  • Calculation errors in division or addition.

Revision tips

  • Practice calculating the mean and median first, as these are foundational steps.
  • Pay close attention to taking the absolute value of each deviation.
  • Double-check all arithmetic calculations, especially sums and divisions.
  • Work through examples for both Mean Deviation about the Mean and Median to understand the differences.
  • Use the provided solutions to verify your own step-by-step calculations.

Practice MCQs

Q1. What is the first step in calculating the Mean Deviation about the Mean?

Q2. For the data {4, 7, 8, 9, 10, 12, 13, 17}, what is the Mean Deviation about the Mean?

Q3. Which measure of central tendency is used for Mean Deviation about the Median?

Q4. What is the median for the data {10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18}?

Q5. The absolute values of deviations are always:

Frequently asked questions

What is Chapter 15 of CBSE Class 11 Mathematics about?

Chapter 15 of CBSE Class 11 Mathematics covers Statistics, focusing on measures of dispersion, specifically the Mean Deviation about the Mean and Mean Deviation about the Median.

How do these NCERT Solutions help students?

These solutions provide clear, step-by-step explanations for calculating Mean Deviation, helping students understand the concepts and methods required for their exams.

What is the difference between Mean Deviation about the Mean and Mean Deviation about the Median?

The primary difference lies in the central point used for calculating deviations. Mean Deviation about the Mean uses the arithmetic mean, while Mean Deviation about the Median uses the median of the data.

Why is it important to calculate the absolute value of deviations?

Calculating the absolute value ensures that all deviations contribute positively to the total dispersion, regardless of whether the data point is above or below the central tendency.

Are the questions in the source document the same as in the NCERT textbook?

Yes, the questions are preserved exactly as they appear in the source document, ensuring alignment with the NCERT curriculum.

What is the first step to find the Mean Deviation about the Median?

The first step is to arrange the given data in ascending order and then find the median of the dataset.

Important links

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