8 Most VSAQ’s of Measures of Dispersion Chapter in Inter 2nd Year Maths-2A (TS/AP)

Table of Contents

2 Marks

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 x_i – 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 $$M = (11+1)/2$$ = 6^th 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 $$\sigma = \sqrt{24.25} = 4.92$$