Calculating Standard Deviation for a Set of Data with Frequencies

To calculate the standard deviation for a set of data with frequencies, we need to follow a few steps.

Step 1: Calculate the Mean

The first step is to calculate the mean of the data set. We can do this by using the following formula:

Mean = (Σf*x) / Σf

Where Σf is the sum of the frequencies, x is the value of the data point, and Σf*x is the sum of the products of the frequencies and the data points.

Using the values given in the problem, we can calculate the mean as follows:

Mean = (10*5 + 14*8 + 8*10 + 5*3 + 5*16) / (10 + 14 + 8 + 5 + 5) = 7.95

Step 2: Calculate the Deviations

The next step is to calculate the deviations of each data point from the mean. We can do this by subtracting the mean from each data point.

Using the values given in the problem, we can calculate the deviations as follows:

Deviation from 5 = 5 - 7.95 = -2.95
Deviation from 8 = 8 - 7.95 = 0.05
Deviation from 10 = 10 - 7.95 = 2.05
Deviation from 3 = 3 - 7.95 = -4.95
Deviation from 16 = 16 - 7.95 = 8.05

Step 3: Square the Deviations

The next step is to square each deviation.

Using the values given in the problem, we can square the deviations as follows:

(-2.95)^2 = 8.7025
(0.05)^2 = 0.0025
(2.05)^2 = 4.2025
(-4.95)^2 = 24.5025
(8.05)^2 = 64.8025

Step 4: Calculate the Sum of the Squared Deviations

The next step is to calculate the sum of the squared deviations.

Using the values given in the problem, we can calculate the sum of the squared deviations as follows:

Σ(x - x̄)^2 = 8.7025 + 0.0025 + 4.2025 + 24.5025 + 64.8025 = 102.215

Step 5: Calculate the Variance

The next step is to calculate the variance. We can do this by dividing the sum of the squared deviations by the total frequency minus one.

Using the values given in the problem, we can calculate the variance as follows:

Variance = Σ(x - x̄)^2 / (Σf - 1) = 102.215 / (42 - 1) = 2.504

Step 6: Calculate the Standard Deviation

The final step is to calculate the standard deviation. We can do this by taking the square root of the variance.

Using the values given in the problem, we can calculate the standard deviation as follows