Variance and standard deviation — calculate for discrete frequency distribution

Question

The marks of 30 students in a test are given as a frequency distribution:

Marks ( $x_i$ )	10	20	30	40	50
Frequency ( $f_i$ )	4	6	10	7	3

Calculate the variance and standard deviation.

(NCERT Class 11, Statistics)

Solution — Step by Step

\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

$x_i$	$f_i$	$f_i x_i$
10	4	40
20	6	120
30	10	300
40	7	280
50	3	150
Total	30	890

\bar{x} = \frac{890}{30} = 29.67

$x_i$	$f_i$	$d_i = x_i - \bar{x}$	$d_i^2$	$f_i d_i^2$
10	4	$-19.67$	386.91	1547.64
20	6	$-9.67$	93.51	561.06
30	10	$0.33$	0.11	1.10
40	7	$10.33$	106.71	746.97
50	3	$20.33$	413.31	1239.93
Total	30			4096.70

\sigma^2 = \frac{\sum f_i d_i^2}{\sum f_i} = \frac{4096.70}{30} = \mathbf{136.56}

\sigma = \sqrt{\sigma^2} = \sqrt{136.56} = \mathbf{11.69}

The standard deviation tells us that marks are spread about 11.69 units around the mean of 29.67.

Why This Works

Variance measures the average of squared deviations from the mean. We square the deviations because positive and negative deviations would cancel out if we simply averaged them. Squaring ensures all terms are positive.

Standard deviation is the square root of variance — it brings the measure back to the original units (marks, in this case). A large standard deviation means data is widely spread; a small one means data is tightly clustered around the mean.

The frequency $f_i$ acts as a weight — values that appear more often contribute more to the variance. This is why we multiply each $d_i^2$ by $f_i$ before summing.

Alternative Method

Use the shortcut formula to avoid calculating deviations:

\sigma^2 = \frac{\sum f_i x_i^2}{\sum f_i} - \left(\frac{\sum f_i x_i}{\sum f_i}\right)^2 = \frac{\sum f_i x_i^2}{N} - \bar{x}^2

Compute $\sum f_i x_i^2 = 4(100) + 6(400) + 10(900) + 7(1600) + 3(2500) = 400 + 2400 + 9000 + 11200 + 7500 = 30500$

\sigma^2 = \frac{30500}{30} - (29.67)^2 = 1016.67 - 880.31 = 136.36

(Small difference due to rounding in $\bar{x}$ .)

The shortcut formula $\sigma^2 = \overline{x^2} - \bar{x}^2$ (mean of squares minus square of mean) is faster for calculations. For JEE, also know the step deviation method where you use $u_i = (x_i - A)/h$ with assumed mean $A$ and class width $h$ to simplify the arithmetic further.

Common Mistake

Students often forget to divide by $N$ (total frequency) at the end and instead divide by the number of distinct values (5 in this case). The divisor for variance is always $N = \sum f_i$ (total number of observations), not the number of classes. With 30 students, you divide by 30, not by 5.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

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