GSEB Class 11 Statistics Solutions Chapter 5 Skewness of Frequency Distribution Exercise 5.2

Gujarat Board Textbook Solutions Class 11 Statistics Chapter 5 Skewness of Frequency Distribution Ex 5.2

Question 1. From the following frequency distribution of different youngsters exercising in a gymnasium, find coefficient of skewness by Bowley's method and state the type of skewness:

Age (years)	25	17	20	18	26	22	28	23
No. of youngsters	22	4	19	11	7	9	3	8

Answer: To begin, we arrange the observed ages in ascending order and prepare the frequency distribution and cumulative frequency (cf) table as shown below:

Age (years)	No. of youngsters (f)	Cumulative frequency (cf)
17	4	4
18	11	15
20	19	34
22	9	43
23	8	51
25	22	73
26	7	80
28	3	83
Total	n = 83

First Quartile:

\[ Q_1 = \text{value of } \left(\frac{n+1}{4}\right)\text{th observation} \] \[ = \text{value of } \left(\frac{83+1}{4}\right)\text{th observation} \] \[ = \text{value of } 21\text{th observation} \] Referring to the cumulative frequency (cf) column, \( Q_1 = 20 \) years.

Median:

\[ M = \text{value of } \left(\frac{n+1}{2}\right)\text{th observation} \] \[ = \text{value of } \left(\frac{83+1}{2}\right)\text{th observation} \] \[ = \text{value of } 42\text{th observation} \] Referring to the cumulative frequency (cf) column, \( M = 22 \) years.

Third Quartile:

\[ Q_3 = \text{value of } 3\left(\frac{n+1}{4}\right)\text{th observation} \] \[ = \text{value of } 3(21)\text{th observation} \] \[ = \text{value of } 63\text{rd observation} \] Referring to the cumulative frequency (cf) column, \( Q_3 = 25 \) years.

Coefficient of skewness by Bowley's Method:

\[ j = \frac{Q_3+Q_1-2M}{Q_3-Q_1} \] \[ = \frac{25+20-2(22)}{25-20} \] \[ = \frac{45-44}{5} \] \[ = \frac{1}{5} \] \[ = 0.2 \]

Type of skewness:
Since \( j = 0.2 \), which is a positive value, the given distribution is positively skewed.

In simple words: First, we organize the data by age and count. Then, we find the first quartile (Q1), median (M), and third quartile (Q3) from the cumulative frequencies. Finally, we use Bowley's formula for the coefficient of skewness. Since the result is positive, the data is skewed to the right.

🎯 Exam Tip: Remember to arrange data in ascending order for quartiles and median calculations. Clearly show each step of the Bowley's coefficient formula to ensure full marks. Interpreting the sign of the coefficient (positive/negative) for skewness type is crucial.

Question 2. The frequency distribution of paid up share capital out of the issued share capital for 31 manufacturing companies is as follows. Find skewness and its coefficient by Bowley's method and state the type of skewness.

Paid up share capital (lakh Rs.)	Less than 100	Less than 300	Less than 500	Less than 700	Less than 900	Less than 1100	Less than 1300
No. of companies	0	6	16	19	23	27	31

Answer: The provided data represents a 'less than' type cumulative frequency distribution. The consistent difference between consecutive upper boundary points is 200, indicating a class length (c) of 200. To calculate the coefficient of skewness using Bowley's method, we first need to convert this into an original frequency distribution.

Paid up share capital (lakh Rs.)	No. of companies (f)	Cumulative frequency (cf)
Less than 100	0	0
100-300	6	6
300-500	10	16
500-700	3	19
700-900	4	23
900-1100	4	27
1100-1300	4	31
Total	n = 31

First Quartile:

\( Q_1 \) class = class that includes \( \frac{n}{4} \)th observation

\( \implies \) class that includes \( \frac{31}{4} \)th observation
\( \implies \) class that includes 7.75th observation Referring to the cumulative frequency (cf) column, the \( Q_1 \) class is 300-500.

Now, \( Q_1 = L + \frac{\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 300 \), \( \frac{n}{4} = 7.75 \), \( cf = 6 \), \( f = 10 \) and \( c = 200 \) in the formula,
\( Q_1 = 300 + \frac{7.75-6}{10} \times 200 \)
\( = 300 + (1.75 \times 20) \)
\( = 300 + 35 \)
\( = 335 \) lakh

Median:

\( M \) class = class that includes \( \left(\frac{n}{2}\right) \)th observation

\( \implies \) class that includes \( \left(\frac{31}{2}\right) \)th observation
\( \implies \) class that includes 15.5th observation Referring to the cumulative frequency (cf) column, the \( M \) class is 300-500.

Now, \( M = L + \frac{\frac{n}{2}-c f}{f} \times c \)

Putting \( L = 300 \), \( \frac{n}{2} = 15.5 \), \( cf = 6 \), \( f = 10 \) and \( c = 200 \) in the formula,
\( M = 300 + \frac{15.5-6}{10} \times 200 \)
\( = 300 + (9.5 \times 20) \)
\( = 300 + 190 \)
\( = 490 \) lakh

Third Quartile:

\( Q_3 \) class = class that includes \( 3\left(\frac{n}{4}\right) \)th observation

\( \implies \) class that includes \( 3(7.75) \)th observation
\( \implies \) class that includes 23.25th observation Referring to the cumulative frequency (cf) column, the \( Q_3 \) class is 900-1100.

Now, \( Q_3 = L + \frac{3\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 900 \), \( 3\left(\frac{n}{4}\right) = 23.25 \), \( cf = 23 \), \( f = 4 \) and \( c = 200 \) in the formula,
\( Q_3 = 900 + \frac{23.25-23}{4} \times 200 \)
\( = 900 + (0.25 \times 50) \)
\( = 900 + 12.5 \)
\( = 912.5 \) lakh

Bowley's method:

Skewness:
\( S_k = Q_3 + Q_1 - 2M \)

\( = 912.5 + 335 - 2(490) \)
\( = 1247.5 - 980 \)
\( = 267.5 \) lakh

Coefficient of skewness:

\[ j = \frac{Q_3+Q_1-2M}{Q_3-Q_1} \] \[ = \frac{S_k}{Q_3-Q_1} \] \[ = \frac{267.5}{912.5-335} = \frac{267.5}{577.5} \] \[ = 0.46 \]

Type of skewness:
Since \( j = 0.46 \), which is a positive value, the frequency distribution is positively skewed.

In simple words: First, convert the 'less than' cumulative frequency to a regular frequency distribution. Then, calculate Q1, M, and Q3 using the appropriate formulas for grouped data. Finally, apply Bowley's coefficient formula. A positive result indicates positive skewness.

🎯 Exam Tip: Converting 'less than' cumulative frequency into a standard frequency distribution is the crucial first step. Ensure correct identification of class boundaries, lower limit (L), frequency (f), cumulative frequency of preceding class (cf), and class length (c) for each quartile and median calculation.

Question 3. The distribution of sales (In thousand tons) of 400 companies during the year 2014-15 Is as follows. Find skewness and its coefficient from these data and state the type of skewness.

Sales (thousand tons)	Less than 20	20-40	40-50	50-75	75-90	90-120	120 and above
No. of companies	30	70	125	100	40	20	15

Answer: We first construct the cumulative frequency distribution table based on the given sales data:

Sale (thousand tons)	No. of companies (f)	Cumulative frequency (cf)
Less than 20	30	30
20-40	70	100
40-50	125	225
50-75	100	325
75-90	40	365
90-120	20	385
120 and more	15	400
Total	n = 400

First Quartile:

\( Q_1 \) class = class that includes \( \frac{n}{4} \)th observation

\( \implies \) class that includes \( \frac{400}{4} \)th observation
\( \implies \) class that includes 100th observation Referring to the cumulative frequency (cf) column, the \( Q_1 \) class is 20-40.

Now, \( Q_1 = L + \frac{\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 20 \), \( \frac{n}{4} = 100 \), \( cf = 30 \), \( f = 70 \) and \( c = 20 \) in the formula,
\( Q_1 = 20 + \frac{100-30}{70} \times 20 \)
\( = 20 + \frac{70 \times 20}{70} \)
\( = 20 + 20 \)
\( = 40 \) thousand tons

Median:

\( M \) class = class that includes \( \left(\frac{n}{2}\right) \)th observation

\( \implies \) class that includes \( \left(\frac{400}{2}\right) \)th observation
\( \implies \) class that includes 200th observation Referring to the cumulative frequency (cf) column, the \( M \) class is 40-50.

Now, \( M = L + \frac{\frac{n}{2}-c f}{f} \times c \)

Putting \( L = 40 \), \( \frac{n}{2} = 200 \), \( cf = 100 \), \( f = 125 \) and \( c = 10 \) in the formula,
\( M = 40 + \frac{200-100}{125} \times 10 \)
\( = 40 + \frac{100 \times 10}{125} \)
\( = 40 + 8 \)
\( = 48 \) thousand tons

Third Quartile:

\( Q_3 \) class = class that includes \( 3\left(\frac{n}{4}\right) \)th observation

\( \implies \) class that includes \( 3(100) \)th observation
\( \implies \) class that includes 300th observation Referring to the cumulative frequency (cf) column, the \( Q_3 \) class is 50-75.

Now, \( Q_3 = L + \frac{3\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 50 \), \( 3\left(\frac{n}{4}\right) = 300 \), \( cf = 225 \), \( f = 100 \) and \( c = 25 \) in the formula (The original class length for 50-75 is 25),
\( Q_3 = 50 + \frac{300-225}{100} \times 25 \)
\( = 50 + \frac{75}{100} \times 25 \)
\( = 50 + 18.75 \)
\( = 68.75 \) thousand tons

Bowley's method:

Skewness:
\( S_k = Q_3 + Q_1 - 2M \)

\( = 68.75 + 40 - 2(48) \)
\( = 108.75 - 96 \)
\( = 12.75 \) thousand tons

Coefficient of skewness:

\[ j = \frac{Q_3+Q_1-2M}{Q_3-Q_1} \] \[ = \frac{S_k}{Q_3-Q_1} \] \[ = \frac{12.75}{68.75-40} = \frac{12.75}{28.75} \] \[ = 0.44 \]

Type of skewness:
Since \( j = 0.44 \), which is a positive value, the frequency distribution is positively skewed.

In simple words: Create a cumulative frequency table from the given sales data. Calculate Q1, M, and Q3 using the grouped data formulas. Then, compute Bowley's coefficient of skewness. A positive coefficient indicates a positive skew.

🎯 Exam Tip: Pay close attention to varying class lengths when calculating Q1, M, and Q3 for grouped data. Double-check the values of L, cf, f, and c specific to each respective class interval to avoid calculation errors.

Question 4. The commission paid on insurance policy amount to agents in a branch of an Insurance company during a month has the following frequency distribution. Find the coefficient of skewness by Bowley's method.

Commission paid (thousand Rs.)	10-12	12-14	14-16	16-18	18-20	20-22	22-24	24-26	26-28	28-30
No. of companies	4	10	16	29	52	80	32	23	17	1

Answer: We construct the cumulative frequency distribution table from the given data:

Commission paid (thousand Rs.)	No. of agents (f)	Cumulative frequency (cf)
10-12	4	4
12-14	10	14
14-16	16	30
16-18	29	59
18-20	52	111
20-22	80	191
22-24	32	223
24-26	23	246
26-28	17	263
28-30	1	264
Total	n = 264

First Quartile:

\( Q_1 \) class = class that includes \( \frac{n}{4} \)th observation

\( \implies \) class that includes \( \frac{264}{4} \)th observation
\( \implies \) class that includes 66th observation Referring to the cumulative frequency (cf) column, the \( Q_1 \) class is 18-20.

Now, \( Q_1 = L + \frac{\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 18 \), \( \frac{n}{4} = 66 \), \( cf = 59 \), \( f = 52 \) and \( c = 2 \) in the formula,
\( Q_1 = 18 + \frac{66-59}{52} \times 2 \)
\( = 18 + \frac{7 \times 2}{52} \)
\( = 18 + \frac{14}{52} \)
\( = 18 + 0.27 \)
\( = 18.27 \) thousand

Median:

\( M \) class = class that includes \( \left(\frac{n}{2}\right) \)th observation

\( \implies \) class that includes \( \left(\frac{264}{2}\right) \)th observation
\( \implies \) class that includes 132nd observation Referring to the cumulative frequency (cf) column, the \( M \) class is 20-22.

Now, \( M = L + \frac{\frac{n}{2}-c f}{f} \times c \)

Putting \( L = 20 \), \( \frac{n}{2} = 132 \), \( cf = 111 \), \( f = 80 \) and \( c = 2 \) in the formula,
\( M = 20 + \frac{132-111}{80} \times 2 \)
\( = 20 + \frac{21 \times 2}{80} \)
\( = 20 + \frac{42}{80} \)
\( = 20 + 0.53 \)
\( = 20.53 \) thousand

Third Quartile:

\( Q_3 \) class = class that includes \( 3\left(\frac{n}{4}\right) \)th observation

\( \implies \) class that includes \( 3(66) \)th observation
\( \implies \) class that includes 198th observation Referring to the cumulative frequency (cf) column, the \( Q_3 \) class is 22-24.

Now, \( Q_3 = L + \frac{3\left(\frac{n}{4}\right)-c f}{f} \times c \)

Putting \( L = 22 \), \( 3\left(\frac{n}{4}\right) = 198 \), \( cf = 191 \), \( f = 32 \) and \( c = 2 \) in the formula,
\( Q_3 = 22 + \frac{198-191}{32} \times 2 \)
\( = 22 + \frac{7 \times 2}{32} \)
\( = 22 + \frac{14}{32} \)
\( = 22 + 0.44 \)
\( = 22.44 \) thousand

Bowley's method:

Skewness:
\( S_k = Q_3 + Q_1 - 2M \)

\( = 22.44 + 18.27 - 2(20.53) \)
\( = 40.71 - 41.06 \)
\( = -0.35 \) thousand

Coefficient of skewness:

\[ j = \frac{Q_3+Q_1-2M}{Q_3-Q_1} \] \[ = \frac{S_k}{Q_3-Q_1} \] \[ = \frac{-0.35}{22.44-18.27} \] \[ = \frac{-0.35}{4.17} \] \[ = -0.08 \]
Since \( j = -0.08 \), which is a negative value, the frequency distribution is negatively skewed.In simple words: First, create a cumulative frequency distribution from the given data. Then, calculate the first quartile (Q1), median (M), and third quartile (Q3) for the grouped data. Finally, use these values in Bowley's formula to find the coefficient of skewness. A negative coefficient indicates negative skewness.

🎯 Exam Tip: When dealing with continuous grouped data, accurately identifying the class interval for Q1, M, and Q3 is key. Make sure to use the correct lower limit (L), frequency (f) of the median/quartile class, and cumulative frequency (cf) of the preceding class for precise calculations.