Calculating Standard Deviation

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

Step 1: Calculate the Mean
To calculate the mean, we need to multiply each value by its respective frequency and sum them up. Then divide the sum by the total frequency.

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

Mean = (50 + 112 + 80 + 15 + 80) / 42

Mean = 337 / 42

Mean = 8.02 (rounded to two decimal places)

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

Deviation for 5 = 5 - 8.02 = -3.02
Deviation for 8 = 8 - 8.02 = -0.02
Deviation for 10 = 10 - 8.02 = 1.98
Deviation for 3 = 3 - 8.02 = -5.02
Deviation for 16 = 16 - 8.02 = 7.98

Step 3: Calculate Squared Deviation
Now, we square each deviation to eliminate negative values and give more weight to larger deviations.

Squared Deviation for 5 = (-3.02)^2 = 9.1204
Squared Deviation for 8 = (-0.02)^2 = 0.0004
Squared Deviation for 10 = (1.98)^2 = 3.9204
Squared Deviation for 3 = (-5.02)^2 = 25.2004
Squared Deviation for 16 = (7.98)^2 = 63.5204

Step 4: Calculate Variance
The variance is the average of the squared deviations. We sum up the squared deviations and divide by the total frequency.

Variance = (9.1204*10 + 0.0004*14 + 3.9204*8 + 25.2004*5 + 63.5204*5) / (10 + 14 + 8 + 5 + 5)

Variance = (91.204 + 0.0056 + 31.3632 + 126.002 + 317.602) / 42

Variance = 566.1762 / 42

Variance = 13.4799 (rounded to four decimal places)

Step 5: Calculate Standard Deviation
Finally, we calculate the standard deviation by taking the square root of the variance.

Standard Deviation = √13.4799

Standard Deviation = 3.67 (rounded to two decimal places)

Conclusion
The standard deviation for the given data set is 3.67. Standard deviation is a measure of how spread out the values are from the mean. A higher standard deviation indicates greater variability in the data.