CBSE Class 11 Mathematics Statistics Notes Set 01

Important Notes and Formulas

Introduction

An average or a central value of a statistical series is the value of the variable which describes the characteristics of the entire distribution. The arithmetic mean and the median are the measures of central tendency used in this chapter; the mean deviation, variance and standard deviation are measured about them.

Arithmetic Mean

Arithmetic mean is the most important among the mathematical means. The arithmetic mean is the amount secured by dividing the sum of values of the items in a series by their number.
(1) Simple arithmetic mean in individual series (ungrouped data)
(i) Direct method: If the series in this case be \( x_1, x_2, x_3, \ldots, x_n \), then the arithmetic mean \( \bar{x} \) is given by
\( \bar{x} = \frac{\text{Sum of the series}}{\text{Number of terms}} \), i.e., \( \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} = \frac{1}{n}\sum_{i=1}^{n} x_i \)
(ii) Short cut method: Arithmetic mean \( \bar{x} = A + \frac{\Sigma d}{n} \), where A = assumed mean, d = deviation from assumed mean = x − A, where x is the individual item, Σd = sum of deviations and n = number of items.

(2) Arithmetic mean in a continuous series (grouped data)
(i) Direct method: If the terms of the given series be \( x_1, x_2, \ldots, x_n \) and the corresponding frequencies be \( f_1, f_2, \ldots, f_n \), then the arithmetic mean \( \bar{x} \) is given by
\( \bar{x} = \frac{f_1x_1 + f_2x_2 + \ldots + f_nx_n}{f_1 + f_2 + \ldots + f_n} = \frac{\sum_{i=1}^{n} f_ix_i}{\sum_{i=1}^{n} f_i} \)
(ii) Short cut method: Arithmetic mean \( \bar{x} = A + \frac{\Sigma f(x - A)}{\Sigma f} \), where A = assumed mean, f = frequency and x − A = deviation of each item from the assumed mean.

(3) Properties of arithmetic mean
(i) The algebraic sum of the deviations of a set of values from their arithmetic mean is zero. If \( x_i/f_i \), i = 1, 2, …, n is the frequency distribution, then \( \sum_{i=1}^{n} f_i(x_i - \bar{x}) = 0 \), \( \bar{x} \) being the mean of the distribution.
(ii) The sum of the squares of the deviations of a set of values is minimum when taken about the mean.
(iii) Mean of the composite series: If \( \bar{x}_i \) (i = 1, 2, …, k) are the means of k component series of sizes \( n_i \) (i = 1, 2, …, k) respectively, then the mean \( \bar{x} \) of the composite series obtained on combining the component series is given by
\( \bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2 + \ldots + n_k\bar{x}_k}{n_1 + n_2 + \ldots + n_k} = \frac{\sum_{i=1}^{k} n_i\bar{x}_i}{\sum_{i=1}^{k} n_i} \)

Median

Median is defined as the value of an item or observation above or below which lies an equal number of observations, i.e., the median is the central value of the set of observations provided all the observations are arranged in ascending or descending order.
Calculation of median
(i) Individual series: If the data is raw, arrange it in ascending or descending order. Let n be the number of observations.
If n is odd, Median = value of \( \left(\frac{n+1}{2}\right)^{th} \) item.
If n is even, Median = \( \frac{1}{2}\left[\text{value of } \left(\frac{n}{2}\right)^{th} \text{ item} + \text{value of } \left(\frac{n}{2} + 1\right)^{th} \text{ item}\right] \)
(ii) Discrete series: In this case, we first find the cumulative frequencies of the variables arranged in ascending or descending order and the median is given by Median = \( \left(\frac{n+1}{2}\right)^{th} \) observation, where n is the cumulative frequency (total frequency).
(iii) For grouped or continuous distributions: In this case, the following formula can be used
(a) For series in ascending order, Median = \( l + \frac{\frac{N}{2} - C}{f} \times i \)
where l = lower limit of the median class, f = frequency of the median class, N = the sum of all frequencies, i = the width of the median class, C = the cumulative frequency of the class preceding the median class.
(b) For series in descending order, Median = \( u - \frac{\frac{N}{2} - C}{f} \times i \), where u = upper limit of the median class and \( N = \sum_{i=1}^{n} f_i \).

Measures of Dispersion

The degree to which numerical data tend to spread about an average value is called the dispersion of the data. The measures of dispersion studied here are (1) Range, (2) Mean deviation, (3) Variance and (4) Standard deviation.
(1) Range: It is the difference between the values of the extreme items in a series. Range = \( X_{max} - X_{min} \).
Range is not a measure of central tendency. Range is widely used in statistical series relating to quality control in production.

(2) Mean deviation: The arithmetic average of the deviations (all taken positive) from the mean or median is known as the mean deviation.
(i) Mean deviation from ungrouped data (or individual series)
Mean deviation = \( \frac{\Sigma|x - M|}{n} \)
where |x − M| means the modulus of the deviation of the variate from the mean or median M, and n is the number of terms.
(ii) Mean deviation from a continuous series: Here first of all we find the mean (or median) from which the deviation is to be taken. Then we find the deviation dM = |x − M| of each variate from the M so obtained. Next we multiply these deviations by the corresponding frequency and find the product f·dM and then the sum ΣfdM of these products. Lastly we use the formula
Mean deviation = \( \frac{\Sigma f|x - M|}{n} = \frac{\Sigma f\,dM}{n} \), where n = Σf
Note: The sum of the deviations of the items from the median, ignoring algebraic signs, is less than the sum from any other point. So the mean deviation is least when it is taken about the median.

(3) Standard deviation: Standard deviation (or S.D.) is the square root of the arithmetic mean of the squares of deviations of various values from their arithmetic mean and is generally denoted by σ, read as sigma.
(i) Standard deviation from an individual series
\( \sigma = \sqrt{\frac{\Sigma(x - \bar{x})^2}{N}} \)
where \( \bar{x} \) = the arithmetic mean of the series and N = the total number of observations.
(ii) Standard deviation from a continuous series
\( \sigma = \sqrt{\frac{\Sigma f_i(x_i - \bar{x})^2}{N}} \)
where \( \bar{x} \) = arithmetic mean of the series, \( x_i \) = mid value of the class, \( f_i \) = frequency of the corresponding \( x_i \), N = Σf = the total frequency.
(iii) Short cut method
(a) \( \sigma = \sqrt{\frac{\Sigma fd^2}{N} - \left(\frac{\Sigma fd}{N}\right)^2} \)    (b) \( \sigma = \sqrt{\frac{\Sigma d^2}{N} - \left(\frac{\Sigma d}{N}\right)^2} \)
where d = x − A = deviation from the assumed mean A, f = frequency of the item, N = Σf = sum of frequencies. Formula (a) is used for a frequency distribution and (b) for an individual series.

Variance

The square of the standard deviation is called the variance.
\( \sigma^2 = \frac{1}{N}\Sigma f_i(x_i - \bar{x})^2 = \frac{\Sigma f_ix_i^2}{N} - \left(\frac{\Sigma f_ix_i}{N}\right)^2 \)
Note:
(i) Standard deviation ≤ Range, i.e., variance ≤ (Range)².
(ii) The S.D. of the first n natural numbers is \( \sqrt{\frac{n^2 - 1}{12}} \).

Solved Examples

Example 2. In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class are 72, then what are the average marks of the girls?
(a) 73
(b) 65
(c) 68
(d) 74
Answer:
Ans. (b) 65
Let the average marks of the girl students be x, then
\( 72 = \frac{70 \times 75 + 30 \times x}{100} \) (Number of girls = 100 − 70 = 30)
i.e., \( \frac{7200 - 5250}{30} = x \), ∴ x = 65.

Example 3. If the mean of the set of numbers \( x_1, x_2, x_3, \ldots, x_n \) is \( \bar{x} \), then the mean of the numbers \( x_i + 2i \), 1 ≤ i ≤ n is
(a) \( \bar{x} + 2n \)
(b) \( \bar{x} + n + 1 \)
(c) \( \bar{x} + 2 \)
(d) \( \bar{x} + n \)
Answer:
Ans. (b) \( \bar{x} + n + 1 \)
We know that \( \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \), i.e., \( \sum_{i=1}^{n} x_i = n\bar{x} \)
∴ \( \frac{\sum_{i=1}^{n}(x_i + 2i)}{n} = \frac{\sum x_i + 2\sum i}{n} = \frac{n\bar{x} + 2(1 + 2 + \ldots + n)}{n} \)
\( = \frac{n\bar{x} + 2 \cdot \frac{n(n+1)}{2}}{n} = \bar{x} + (n + 1) \)

Example 4. The average of n numbers \( x_1, x_2, x_3, \ldots, x_n \) is M. If \( x_n \) is replaced by x′, then the new average is
(a) \( M - x_n + x' \)
(b) \( \frac{nM - x_n + x'}{n} \)
(c) \( \frac{(n-1)M + x'}{n} \)
(d) \( \frac{M - x_n + x'}{n} \)
Answer:
Ans. (b) \( \frac{nM - x_n + x'}{n} \)
\( M = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \), i.e., \( nM = x_1 + x_2 + x_3 + \ldots + x_{n-1} + x_n \)
\( nM - x_n = x_1 + x_2 + x_3 + \ldots + x_{n-1} \)
\( \frac{nM - x_n + x'}{n} = \frac{x_1 + x_2 + x_3 + \ldots + x_{n-1} + x'}{n} \)
∴ New average = \( \frac{nM - x_n + x'}{n} \)

Example 5. Mean of 100 items is 49. It was discovered that three items which should have been 60, 70, 80 were wrongly read as 40, 20, 50 respectively. The correct mean is
(a) 48
(b) \( 82\frac{1}{2} \)
(c) 50
(d) 80
Answer:
Ans. (c) 50
Sum of 100 items = 49 × 100 = 4900
Sum of items added = 60 + 70 + 80 = 210
Sum of items replaced = 40 + 20 + 50 = 110
New sum = 4900 + 210 − 110 = 5000
∴ Correct mean = \( \frac{5000}{100} = 50 \)

Example 7. The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set
(a) is increased by 2
(b) is decreased by 2
(c) is two times the original median
(d) remains the same as that of the original set
Answer:
Ans. (d) remains the same as that of the original set
n = 9, then median term = \( \left(\frac{9 + 1}{2}\right)^{th} \) = 5th term. Since only the last four observations are increased by 2,
∵ the median is the 5th observation, which remains unchanged.
∴ There will be no change in the median.

Example 9. A batsman scores runs in 10 innings 38, 70, 48, 34, 42, 55, 63, 46, 54, 44, then the mean deviation about the median is
(a) 8.6
(b) 6.4
(c) 10.6
(d) 9.6
Answer:
Ans. (a) 8.6
Arranging the given data in ascending order, we have 34, 38, 42, 44, 46, 48, 54, 55, 63, 70
Here median M = \( \frac{46 + 48}{2} = 47 \) (∵ n = 10, the median is the mean of the 5th and 6th items)
∴ Mean deviation = \( \frac{\Sigma|x_i - M|}{n} = \frac{\Sigma|x_i - 47|}{10} = \frac{13 + 9 + 5 + 3 + 1 + 1 + 7 + 8 + 16 + 23}{10} = 8.6 \)

Example 10. S.D. of data is 6 when each observation is increased by 1, then the S.D. of new data is
(a) 5
(b) 7
(c) 6
(d) 8
Answer:
Ans. (c) 6
S.D. and variance of data are not changed when each observation is increased (or decreased) by the same constant.

Example 11. In a series of 2n observations, half of them equal a and the remaining half equal −a. If the standard deviation of the observations is 2, then |a| equals
(a) \( \frac{\sqrt{2}}{n} \)
(b) \( \sqrt{2} \)
(c) 2
(d) \( \frac{1}{n} \)
Answer:
Ans. (c) 2
Let a, a, …, n times and −a, −a, …, n times, i.e. mean = 0
and S.D. = \( \sqrt{\frac{n(a - 0)^2 + n(-a - 0)^2}{2n}} \)
\( 2 = \sqrt{\frac{na^2 + na^2}{2n}} = \sqrt{a^2} = |a| \). Hence |a| = 2.

Example 12. If μ is the mean of distribution \( (y_i, f_i) \), then \( \Sigma f_i(y_i - \mu) = \)
(a) M.D.
(b) S.D.
(c) 0
(d) Relative frequency
Answer:
Ans. (c) 0
We have \( \Sigma f_i(y_i - \mu) = \Sigma f_iy_i - \mu\Sigma f_i = \mu\Sigma f_i - \mu\Sigma f_i = 0 \)
\( \left[\because \mu = \frac{\Sigma f_iy_i}{\Sigma f_i}\right] \)

Example 14. In an experiment with 15 observations on x, the following results were available: Σx² = 2830, Σx = 170. One observation that was 20 was found to be wrong and was replaced by the correct value 30. Then the corrected variance is
(a) 78.00
(b) 188.66
(c) 177.33
(d) 8.33
Answer:
Ans. (a) 78.00
Σx = 170, Σx² = 2830
Increase in Σx = 10, then Σx′ = 170 + 10 = 180
Increase in Σx² = 900 − 400 = 500, then Σx′² = 2830 + 500 = 3330
Variance = \( \frac{1}{n}\Sigma x'^2 - \left(\frac{\Sigma x'}{n}\right)^2 = \frac{3330}{15} - \left(\frac{180}{15}\right)^2 = 222 - 144 = 78 \)

Example 15. The S.D. of a variate x is σ. The S.D. of the variate \( \frac{ax + b}{c} \), where a, b, c are constants, is
(a) \( \left(\frac{a}{c}\right)\sigma \)
(b) \( \left|\frac{a}{c}\right|\sigma \)
(c) \( \left(\frac{a^2}{c^2}\right)\sigma \)
(d) None of these
Answer:
Ans. (b) \( \left|\frac{a}{c}\right|\sigma \)
Let \( y = \frac{ax + b}{c} \), i.e., \( y = \frac{a}{c}x + \frac{b}{c} \), i.e. y = Ax + B, where \( A = \frac{a}{c}, B = \frac{b}{c} \)
∴ \( \bar{y} = A\bar{x} + B \)
∴ \( y - \bar{y} = A(x - \bar{x}) \Rightarrow (y - \bar{y})^2 = A^2(x - \bar{x})^2 \)
\( \Rightarrow \Sigma(y - \bar{y})^2 = A^2\Sigma(x - \bar{x})^2 \Rightarrow n\sigma_y^2 = A^2 \cdot n\sigma_x^2 \Rightarrow \sigma_y^2 = A^2\sigma_x^2 \)
\( \Rightarrow \sigma_y = |A|\sigma_x \Rightarrow \sigma_y = \left|\frac{a}{c}\right|\sigma_x \)
Thus, new S.D. = \( \left|\frac{a}{c}\right|\sigma \).