Bowley's coefficient of skewness is 1.00, indicating a perfectly symmetrical distribution.
Bowley's coefficient of skewness (Skb) is a measure of the skewness of a distribution. It is calculated using the following formula:
Skb = (Q3 + Q1 - 2Q2) / (Q3 - Q1)
where Q1, Q2, and Q3 are the first, second, and third quartiles of the distribution, respectively.
To calculate Skb for the given distribution, we first need to find the quartiles. The quartiles are the values that divide the distribution into four equal parts. We can find the quartiles by calculating the cumulative frequencies of the distribution.
The cumulative frequency of a class is the sum of the frequencies of all classes up to and including that class. The cumulative frequencies of the given distribution are shown in the following table:
Class | Frequency | Cumulative frequency
Below 300 | 5 | 5
300-400 | 8 | 13
400-500 | 18 | 31
500-600 | 35 | 66
600-700 | 27 | 93
Above 700 | 7 | 100
The first quartile (Q1) is the median of the lower half of the distribution. The lower half of the distribution is the part below the median. The median is the value that divides the distribution into two equal parts.
To find the median, we first need to find the cumulative frequency of the median class. The median class is the class that contains the median value. The cumulative frequency of the median class must be greater than or equal to N/2, where N is the total number of observations.
In the given distribution, the cumulative frequency of the 500-600 class is 66, which is greater than or equal to 100/2 = 50. Therefore, the median class is the 500-600 class.
The median of the 500-600 class is 550. Therefore, the first quartile (Q1) is 550.
The second quartile (Q2) is the median of the entire distribution. The median of the entire distribution is 550. Therefore, the second quartile (Q2) is 550.
The third quartile (Q3) is the median of the upper half of the distribution. The upper half of the distribution is the part above the median.
To find the third quartile, we first need to find the cumulative frequency of the third quartile class. The third quartile class is the class that contains the third quartile value. The cumulative frequency of the third quartile class must be greater than or equal to 3N/4
In the given distribution, the cumulative frequency of the 600-700 class is 93, which is greater than or equal to 100*3/4 = 75. Therefore, the third quartile class is the 600-700 class.
The median of the 600-700 class is 650. Therefore, the third quartile (Q3) is 650.
Now that we have found the quartiles, we can calculate Skb using the following formula:
Skb = (Q3 + Q1 - 2Q2) / (Q3 - Q1)
Skb = (650 + 550 - 2*550) / (650 - 550)
Skb = 100 / 100
Skb = 1.00
Therefore, Bowley's coefficient of skewness for the given distribution is 1.00. This indicates that the distribution is perfectly symmetrical.