Calculating Standard Deviation by the Actual Mean Method

To calculate the standard deviation by the actual mean method, we need to follow a step-by-step process. Let's go through each step in detail:

Step 1: Organize the Data
The given data is as follows:
Size Frequency
5 2
10 1
15 4
20 3

Step 2: Calculate the Mean
To find the mean, we need to multiply each value by its corresponding frequency, sum up the products, and then divide by the total frequency.

Mean = (5 * 2 + 10 * 1 + 15 * 4 + 20 * 3) / (2 + 1 + 4 + 3)
= (10 + 10 + 60 + 60) / 10
= 140 / 10
= 14

The mean of the given data is 14.

Step 3: Calculate Deviations
Next, we need to calculate the deviation of each value from the mean. Deviation is the difference between each value and the mean.

Deviation = Value - Mean

For the given data, the deviations are as follows:
5 - 14 = -9
10 - 14 = -4
15 - 14 = 1
20 - 14 = 6

Step 4: Square the Deviations
Now, we square each deviation calculated in the previous step.

Squared Deviation = Deviation^2

For the given data, the squared deviations are as follows:
(-9)^2 = 81
(-4)^2 = 16
1^2 = 1
6^2 = 36

Step 5: Calculate the Product of Frequency and Squared Deviation
Next, we multiply the frequency of each value by its corresponding squared deviation.

Product of Frequency and Squared Deviation = Frequency * Squared Deviation

For the given data, the products of frequency and squared deviation are as follows:
2 * 81 = 162
1 * 16 = 16
4 * 1 = 4
3 * 36 = 108

Step 6: Calculate the Sum of the Products of Frequency and Squared Deviation
Now, we sum up the products of frequency and squared deviation.

Sum of the Products of Frequency and Squared Deviation = Σ(Product of Frequency and Squared Deviation)

For the given data, the sum of the products of frequency and squared deviation is:
162 + 16 + 4 + 108 = 290

Step 7: Calculate the Standard Deviation
Finally, we calculate the standard deviation using the formula:

Standard Deviation = √(Sum of the Products of Frequency and Squared Deviation / Total Frequency)

In our case, the standard deviation is:
√(290 / 10) = √29 = 5.385

Therefore, the standard deviation of the given data, calculated by the actual mean method, is approximately 5.385.