GSEB Class 11 Statistics Solutions Chapter 4 Measures of Dispersion Solution Exercise 4.3

GSEB Solutions Class 11 Statistics Chapter 4 Measures Of Dispersion Ex 4.3

 

Question 1. The measurements of height (in centimeters) of 10 soldiers are given below: 160, 175, 158, 165, 170, 166, 173, 176, 163, 168. Find the mean deviation of the heights of the soldiers.

Answer: Given that the number of soldiers, n, is 10. The following table illustrates the calculation steps for mean deviation:

Height (in cm)	\(|x - \bar{x}|\) \( \bar{x} = 167.4 \)
160	7.4
175	7.6
158	9.4
165	2.4
170	2.6
166	1.4
173	5.6
176	8.6
163	4.4
168	0.6
\( \Sigma x = 1674 \)	\( \Sigma |x - \bar{x}| = 50.0 \)

The arithmetic mean \((\bar{x})\) is calculated as the sum of all heights divided by the number of soldiers: \( \bar{x} = \frac{\Sigma x}{n} \)
\( \implies \bar{x} = \frac{1674}{10} \)
\( \implies \bar{x} = 167.4 \text{ cm} \) Subsequently, the mean deviation (MD) of the soldiers' heights is determined by summing the absolute differences from the mean and dividing by n: \( MD = \frac{\Sigma|x - \bar{x}|}{n} \)
\( \implies MD = \frac{50}{10} \)
\( \implies MD = 5 \text{ cm} \)
In simple words: To find the mean deviation, first calculate the average height of the soldiers. Then, find how much each soldier's height differs from this average, ignoring negative signs. Finally, average these differences to get the mean deviation.

🎯 Exam Tip: Always remember to sum the absolute deviations when calculating mean deviation and present the final answer with appropriate units if applicable. Ensure the mean is calculated correctly as it forms the basis for all deviations.

 

Question 2. The distribution of number of ball bearings used in machines of a factory is given below. Calculate the mean deviation and coefficient of mean deviation of number of ball bearings per machine.

No. of ball bearings	No. of machines
2	2
4	2
6	4
8	5
10	3
12	2
14	1
16	1

Answer: To compute the mean deviation and its coefficient, we first determine the mean \((\bar{x})\) using the given frequency distribution. The detailed calculations are shown in the table below:

No. of ball bearing x	No. of machines f	fx	\(|x - \bar{x}|\) \( \bar{x} = 8 \)	\(f|x - \bar{x}|\)
2	2	4	6	12
4	2	8	4	8
6	4	24	2	8
8	5	40	0	0
10	3	30	2	6
12	2	24	4	8
14	1	14	6	6
16	1	16	8	8
Total	\(n = 20\)	\( \Sigma fx = 160 \)	-	\( \Sigma f|x - \bar{x}| = 56 \)

Mean: \( \bar{x} = \frac{\Sigma fx}{n} \)
\( \implies \bar{x} = \frac{160}{20} \)
\( \implies \bar{x} = 8 \text{ ball bearings} \) Next, the mean deviation (MD) is calculated by summing the product of frequencies and absolute deviations from the mean, then dividing by the total number of machines: \( MD = \frac{\Sigma f|x - \bar{x}|}{n} \)
\( \implies MD = \frac{56}{20} \)
\( \implies MD = 2.8 \text{ ball bearings} \) Finally, the coefficient of mean deviation is found by dividing the mean deviation by the mean: \( \text{Coefficient of mean deviation} = \frac{MD}{\bar{x}} \)
\( \implies \text{Coefficient of mean deviation} = \frac{2.8}{8} \)
\( \implies \text{Coefficient of mean deviation} = 0.35 \)
In simple words: To find the mean deviation for grouped data, first calculate the weighted average (mean) of the ball bearings. Then, for each category, find the absolute difference from this mean and multiply it by its frequency. Sum these products and divide by the total frequency to get the mean deviation. The coefficient then compares this deviation to the mean itself.

🎯 Exam Tip: When dealing with frequency distributions, ensure to correctly calculate the mean (\( \bar{x} \)) by considering the frequency (f) for each observation (x). The formula \( \bar{x} = \frac{\Sigma fx}{n} \) is crucial. Also, remember that the coefficient of mean deviation is a relative measure, useful for comparing variability across different datasets.

 

Question 3. Find the mean deviation and the coefficient of mean deviation of the distribution of talk time (in minutes) per call.

Talk time (in minutes)	No. of calls
3	4
5	7
10	6
12	2
15	1

Answer: For this distribution of call talk times, we first establish the mean \((\bar{x})\). The complete calculation process is presented in the table below:

Talk time (in minutes) x	No. of calls f	fx	\(|x - \bar{x}|\) \( \bar{x} = 7.3 \)	\(f|x - \bar{x}|\)
3	4	12	4.3	17.2
5	7	35	2.3	16.1
10	6	60	2.7	16.2
12	2	24	4.7	9.4
15	1	15	7.7	7.7
Total	\(n = 20\)	\( \Sigma fx = 146 \)	-	\( \Sigma f|x - \bar{x}| = 66.6 \)

Mean: \( \bar{x} = \frac{\Sigma fx}{n} \)
\( \implies \bar{x} = \frac{146}{20} \)
\( \implies \bar{x} = 7.3 \text{ minutes} \) Subsequently, the mean deviation (MD) for talk time is computed by aggregating the products of frequencies and the absolute differences from the mean, divided by the total calls: \( MD = \frac{\Sigma f|x - \bar{x}|}{n} \)
\( \implies MD = \frac{66.6}{20} \)
\( \implies MD = 3.33 \text{ minutes} \) The coefficient of mean deviation is then derived by dividing the mean deviation by the mean: \( \text{Coefficient of Mean deviation} = \frac{MD}{\bar{x}} \)
\( \implies \text{Coefficient of Mean deviation} = \frac{3.33}{7.3} \)
\( \implies \text{Coefficient of Mean deviation} = 0.46 \)
In simple words: To calculate mean deviation and its coefficient for talk time, first find the average talk time per call. Then, determine the average absolute difference of each call's duration from this mean. Finally, divide this average difference by the mean to get the coefficient.

🎯 Exam Tip: For continuous data or grouped frequency distributions, ensure you use the mid-value of each class interval for 'x' in your calculations. Pay attention to decimal places for accuracy in final answers, especially for coefficients.

 

Question 4. Find the mean deviation and coefficient of mean deviation of number of TV sets using the following frequency distribution of TV sets sold in last 16 months in a town:

No. of TV sets	No. of months
10-30	1
30-50	4
50-70	6
70-90	4
90-110	1

Answer: Based on the provided frequency distribution for TV set sales, we first calculate the mean \((\bar{x})\). The detailed steps for these calculations are shown in the table below:

No. of TV sets	No. of months f	Mid value x	fx	\(|x - \bar{x}|\) \( \bar{x} = 60 \)	\(f|x - \bar{x}|\)
10-30	1	20	20	40	40
30-50	4	40	160	20	80
50-70	6	60	360	0	0
70-90	4	80	320	20	80
90-110	1	100	100	40	40
Total	\(n = 16\)	-	\( \Sigma fx = 960 \)	-	\( \Sigma f|x - \bar{x}| = 240 \)

Mean: \( \bar{x} = \frac{\Sigma fx}{n} \)
\( \implies \bar{x} = \frac{960}{16} \)
\( \implies \bar{x} = 60 \text{ TV sets} \) The mean deviation (MD) for the monthly sales of TV sets is then found by summing the products of frequencies and the absolute deviations from the mean, divided by n: \( MD = \frac{\Sigma f|x - \bar{x}|}{n} \)
\( \implies MD = \frac{240}{16} \)
\( \implies MD = 15 \text{ TV sets} \) Lastly, the coefficient of mean deviation is calculated as the ratio of MD to the mean: \( \text{Coefficient of mean deviation} = \frac{MD}{\bar{x}} \)
\( \implies \text{Coefficient of mean deviation} = \frac{15}{60} \)
\( \implies \text{Coefficient of mean deviation} = 0.25 \)
In simple words: For grouped data like TV set sales, first find the midpoint of each range. Use these midpoints and frequencies to calculate the average number of TV sets sold (mean). Then, find the average absolute difference of sales from this mean, and finally, divide this average difference by the mean to get the coefficient.

🎯 Exam Tip: When working with class intervals, accurately calculating the mid-value (x) is critical. A common mistake is using the lower or upper limit instead of the midpoint. The steps for calculating mean, mean deviation, and its coefficient should be followed systematically for full marks.

 

Question 5. There are 50 boxes containing different number of units of an item in a factory. Find the mean deviation of number of units per box using the following distribution of the units:

No. of units	No. of boxes
0-10	6
10-20	5
20-30	8
30-40	15
40-50	7
50-60	6
60-70	3

Answer: For the given distribution of units per box, we first compute the mean \((\bar{x})\). The comprehensive calculations are presented in the table below:

No. of units	No. of boxes f	Mid value x	fx	\(|x - \bar{x}|\) \( \bar{x} = 33.4 \)	\(f|x - \bar{x}|\)
0-10	6	5	30	28.4	170.4
10-20	5	15	75	18.4	92.0
20-30	8	25	200	8.4	67.2
30-40	15	35	525	1.6	24.0
40-50	7	45	315	11.6	81.2
50-60	6	55	330	21.6	129.6
60-70	3	65	195	31.6	94.8
Total	\(n = 50\)	-	\( \Sigma fx = 1670 \)	-	\( \Sigma f|x - \bar{x}| = 659.2 \)

Mean: \( \bar{x} = \frac{\Sigma fx}{n} \)
\( \implies \bar{x} = \frac{1670}{50} \)
\( \implies \bar{x} = 33.4 \text{ units} \) Subsequently, the mean deviation (MD) for the number of units per box is determined by summing the products of frequencies and the absolute differences from the mean, then dividing by the total number of boxes: \( MD = \frac{\Sigma f|x - \bar{x}|}{n} \)
\( \implies MD = \frac{659.2}{50} \)
\( \implies MD = 13.18 \text{ units} \)
In simple words: To calculate the mean deviation for this factory data, first find the average number of units per box using midpoints and frequencies. Then, calculate how much each box's unit count deviates from this average, taking the absolute value. Finally, average these absolute deviations to get the mean deviation.

🎯 Exam Tip: Remember to use the correct mid-value for each class interval when calculating \( \Sigma fx \) and \( \Sigma f|x - \bar{x}| \). Accuracy in arithmetic, especially with decimals, is vital for correct mean deviation results.