Measures Of Dispersion (VSAQs)

Maths-2A | 8. Measures Of Dispersion – VSAQs:
Welcome to VSAQs in Chapter 8: Measures Of Dispersion. This page contains the most important VSAQs in this chapter. Aim to secure top marks in your exams by understanding these clear and straightforward Very Short Answer Questions.

VSAQ-1 : Find the mean deviation about mean for the data 3,6,10,4,9,10

Given data: $$3, 6, 10, 4, 9, 10$$

Here $$n = 6$$

Mean $$x = \frac{3 + 6 + 10 + 4 + 9 + 10}{6} = \frac{42}{6} = 7$$

Deviations from the mean:

$$3 – 7 = -4$$ $$6 – 7 = -1$$ $$10 – 7 = 3$$ $$4 – 7 = -3$$ $$9 – 7 = 2$$ $$10 – 7 = 3$$

Absolute values of these deviations : $$4, 1, 3, 3, 2, 3$$

M.D from Mean is $$M.D = \frac{\sum |x_i – x|}{6}$$

$$= \frac{4 + 1 + 3 + 3 + 2 + 3}{6} = \frac{16}{6} = 2.67$$

VSAQ-2 : Find the mean deviation about mean for the data 6,7,10,12,13,4,12,16

Given data: $$6, 7, 10, 12, 13, 4, 12, 16$$

Here $$n = 8$$

Mean $$x = \frac{6 + 7 + 10 + 12 + 13 + 4 + 12 + 16}{8} = \frac{80}{8} = 10$$

Deviation from the mean:

$$6 – 10 = -4$$ $$7 – 10 = -3$$ $$10 – 10 = 0$$ $$12 – 10 = 2$$ $$13 – 10 = 3$$ $$4 – 10 = -6$$ $$12 – 10 = 2$$ $$16 – 10 = 6$$

Absolute values of these deviations: $$4, 3, 0, 2, 3, 6, 2, 6$$

M.D from Mean is $$M.D = \frac{\sum |x_i – x|}{8}$$

$$= \frac{4 + 3 + 0 + 2 + 3 + 6 + 2 + 6}{8} = \frac{26}{8} = 3.25$$

VSAQ-3 : Find the mean deviation about median for the data 4,6,9,3,10,13,2

Given data: $$4, 6, 9, 3, 10, 13, 2$$

Its ascending order: $$2, 3, 4, 6, 9, 10, 13$$

Number of observations n = 7 is odd

Median is the middle most term M = 6

Deviations from the median :

$$2 – 6 = -4$$ $$3 – 6 = -3$$ $$4 – 6 = -2$$ $$6 – 6 = 0$$ $$9 – 6 = 3$$ $$10 – 6 = 4$$ $$13 – 6 = 7$$

Absolute values of these deviations: $$4, 3, 2, 0, 3, 4, 7$$

M.D from Median is $$M.D = \frac{\sum |x_i – M|}{7}$$

$$= \frac{4 + 3 + 2 + 0 + 3 + 4 + 7}{7} = \frac{23}{7} = 3.29$$

VSAQ-4 : Find the mean deviation about median for the data 6,7,10,12,13,4,12,16

Given data: $$6, 7, 10, 12, 13, 4, 12, 16$$

Its ascending order: $$4, 6, 7, 10, 12, 12, 13, 16$$

Number of observations n = 8 is even

Median is $$M = \frac{10 + 12}{2} = 11$$

Deviations from the median:

$$11 – 4 = 7$$ $$11 – 6 = 5$$ $$11 – 7 = 4$$ $$11 – 10 = 1$$ $$11 – 12 = -1$$ $$11 – 12 = -1$$ $$11 – 13 = -2$$ $$11 – 16 = -5$$

Absolute values of these deviations: $$7, 5, 4, 1, 1, 1, 2, 5$$

M.D from Median is $$MD = \frac{\sum |x_i – M|}{8}$$

$$= \frac{7 + 5 + 4 + 1 + 1 + 1 + 2 + 5}{8} = \frac{26}{8} = 3.25$$

VSAQ-5 : Find the mean deviation about mean for the data 38,70,48,40,42,55,63,46,54,44

The given ungrouped data is $$38, 70, 48, 40, 42, 55, 63, 46, 54, 44$$

Here $$n = 10$$

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

The deviation of the observations from the mean xi – x are

$$38 – 50 = -12$$ $$70 – 50 = 20$$ $$48 – 50 = -2$$ $$40 – 50 = -10$$ $$42 – 50 = -8$$ $$55 – 50 = 5$$ $$63 – 50 = 13$$ $$46 – 50 = -4$$ $$54 – 50 = 4$$ $$44 – 50 = -6$$

Hence the absolute values of the deviations are $$12, 20, 2, 10, 8, 5, 13, 4, 4, 6$$

The mean deviation about mean is $$M.D = \frac{\sum |x_i – x|}{10}$$

$$= \frac{12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6}{10} = \frac{84}{10} = 8.4$$

VSAQ-6 : Find the mean deviation about median for the following data 13,17,16,11,13,10,16,11,18,12,17

The ascending order of the observations is $$10, 11, 11, 12, 13, 13, 16, 16, 17, 17, 18$$

The number of observations = 11(odd)

Median of the given data is $$M = (11+1)/2$$ = 6th item = 13

The deviations of observations from the median are

$$10 – 13 = -3$$ $$11 – 13 = -2$$ $$12 – 13 = -1$$ $$13 – 13 = 0$$ $$16 – 13 = 3$$ $$17 – 13 = 4$$ $$18 – 13 = 5$$

Hence the absolute values of the deviations are $$3, 2, 2, 1, 0, 0, 3, 3, 4, 4, 5$$

The mean deviation from the median is $$M.D = \frac{\sum |x_i – M|}{11}$$

$$= \frac{3 + 2 + 2 + 1 + 0 + 0 + 3 + 3 + 4 + 4 + 5}{11} = \frac{27}{11} = 2.45$$

VSAQ-7 : Find the variance for the discrete data 6,7,10,12,13,4,8,12

Mean $$x = \frac{\sum x_i}{n}$$

$$= \frac{6 + 7 + 10 + 12 + 13 + 4 + 8 + 12}{8} = \frac{72}{8} = 9$$

Deviations from the mean:

$$6 – 9 = -3$$ $$7 – 9 = -2$$ $$10 – 9 = 1$$ $$12 – 9 = 3$$ $$13 – 9 = 4$$ $$4 – 9 = -5$$ $$8 – 9 = -1$$ $$12 – 9 = 3$$

Absolute values of these deviations: $$3, 2, 1, 3, 4, 5, 1, 3$$

Variance $$\sigma^2 = \frac{\sum (x_i – x)^2}{n}$$

$$= \frac{3^2 + 2^2 + 1^2 + 3^2 + 4^2 + 5^2 + 1^2 + 3^2}{8}$$

$$= \frac{9 + 4 + 1 + 9 + 16 + 25 + 1 + 9}{8} = \frac{74}{8} = 9.25$$

VSAQ-8 : Find the variance and standard deviation for discrete data: 5,12,3,18,6,8,2,10

Mean $$x = \frac{\sum x_i}{n}$$

$$= \frac{5 + 12 + 3 + 18 + 6 + 8 + 2 + 10}{8} = \frac{64}{8} = 8$$

Deviations from the mean:

$$5 – 8 = -3$$ $$12 – 8 = 4$$ $$3 – 8 = -5$$ $$18 – 8 = 10$$ $$6 – 8 = -2$$ $$8 – 8 = 0$$ $$2 – 8 = -6$$ $$10 – 8 = 2$$

Absolute values of these deviations: 3,4,5,10,2,0,6,2

Variance $$\sigma^2 = \frac{\sum (x_i – x)^2}{n}$$

$$= \frac{3^2 + 4^2 + 5^2 + 10^2 + 2^2 + 0^2 + 6^2 + 2^2}{8}$$

$$= \frac{9 + 16 + 25 + 100 + 4 + 0 + 36 + 4}{8} = \frac{194}{8} = 24.25$$

Standard deviation $$\sigma = \sqrt{24.25} = 4.92$$