Calculating Standard Deviation by the Actual Mean Method

To calculate the standard deviation using the actual mean method, we need to follow a few steps. Let's use the data provided: Size Frequency 5 2 10 1 15 4 20 3.

Step 1: Calculate the mean
To begin, we need to calculate the mean of the data set. The mean is the average of all the values. We can calculate it by summing up all the values and dividing by the total number of values.

Using the given data, we can calculate the mean as follows:

(5 * 2) + (10 * 1) + (15 * 4) + (20 * 3) = 10 + 10 + 60 + 60 = 140

Total frequency = 2 + 1 + 4 + 3 = 10

Mean = 140 / 10 = 14

Step 2: Calculate the deviation from the mean
Next, we need to calculate the deviation of each value from the mean. The deviation is the difference between each value and the mean.

Using the mean calculated in step 1, we can calculate the deviation as follows:

For the value 5: 5 - 14 = -9
For the value 10: 10 - 14 = -4
For the value 15: 15 - 14 = 1
For the value 20: 20 - 14 = 6

Step 3: Square the deviations
After calculating the deviation for each value, we need to square each deviation. This is done to eliminate negative values and emphasize the differences between the values and the mean.

Squaring the deviations calculated in step 2, we get:

For the value 5: (-9)^2 = 81
For the value 10: (-4)^2 = 16
For the value 15: 1^2 = 1
For the value 20: 6^2 = 36

Step 4: Calculate the sum of squared deviations
Next, we need to sum up all the squared deviations calculated in step 3.

Sum of squared deviations = 81 + 16 + 1 + 36 = 134

Step 5: Calculate the variance
To calculate the variance, we need to divide the sum of squared deviations by the total frequency.

Variance = Sum of squared deviations / Total frequency = 134 / 10 = 13.4

Step 6: Calculate the standard deviation
Finally, to calculate the standard deviation, we need to take the square root of the variance.

Standard deviation = √(Variance) = √(13.4) ≈ 3.66

Conclusion
The standard deviation of the given data set, calculated using the actual mean method, is approximately 3.66. This value represents the average amount of deviation from the mean in the data set.