Calculate the sample standard deviation and sample variance for the following frequency distribution of hourly wages for a sample of pharmacy assistants. Lower Bound Upper Bound
6.51 8.50
50 8.51
10.50 18
10.51 12.50
42 12.51
14.50 20
14.51 16.50

(a) standard deviation = σ = 4.9996

(b) variance = σ² = 24.996

Given frequency table (find attached as Table 1);

(a) To find the sample standard deviation and sample variance, follow these steps;

i. Calculate the mid-point c for each group by using the mid-point formula;

c = (lower bound + upper bound) / 2

=> c = (6.51 + 8.50) / 2 = 7.505

=> c = (8.51 + 10.50) / 2 = 9.505

=> c = (10.51 + 12.50) / 2 = 11.505

=> c = (12.51 + 14.50) / 2 = 13.505

=> c = (14.51 + 16.50) / 2 = 15.505

So the new table becomes (find attached as Table 2);

ii. Calculate the total number of samples (n) which is the sum of all the frequencies.

n = 50+18+42+20+46

n = 176

iii. Calculate the mean (M)

This is done by first multiplying the midpoints by the corresponding frequencies and then dividing the result by the total number of samples (n).

M = [(7.505 x 50) + (9.505 x 18) + (11.505 x 42) + (13.505 x 20) + (15.505 x 46)] / 176

M = [375.25 + 171.09 + 483.21 + 270.1 + 713.23] / 176

M = [2012.88] / 176

M = 11.44

iv. Find the variance (σ²);

The variance is calculated using the following formula

σ² = [Σ(f x c²) - (n x M²)] / (n - 1)                ------------(i)

Where;

f = frequency of each boundary data point

=>  Let's first calculate Σ(f x c²).

This is done by finding the sum of the product of the frequency (f) of each boundary point and the square of their corresponding mid-points(c)

Σ(f x c²) = [(50 x 7.505²) + (18 x 9.505²) + (42 x 11.505²) + (20 x 13.505²) + (46 x 15.505²)]

Σ(f x c²) = [(2816.25125) + (1626.21045) + (5559.33105) + (3647.7005) + (11058.63115)]

Σ(f x c²) = 24708.1244

=> Now calculate (n x M²)

n x M² = 176 x 11.44²

n x M² = 23033.7536

=> Now substitute these values into equation (i) to calculate the variance

σ² = [Σ(f x c²) - (n x M²)] / (n - 1)

σ² = [24708.1244 - 23033.7536] / (176 - 1)

σ² = [4374.3708] / (175)

σ² = 24.996

Therefore, the variance is 24.996

v. Find the standard deviation (σ)

The standard deviation is the square root of the variance. i.e

σ = √σ²

σ = √24.996

σ = 4.9996

Therefore, the standard deviation is 4.9996

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