Given the following frequency distribution, find the mean, variance, and standard deviation.


Class Frequency
31-33 18
34-36 25
37-39 22
40-42 9
43-45 20

Find the mean first = sum of scores/number of scores (Use midpoint of classes as the values of the scores.)

Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.

Standard deviation = square root of variance

I'll let you do the calculations.

JAKIAM

To calculate the mean, variance, and standard deviation, we need to find the midpoint of each class interval and create a table with three columns: the midpoint, frequency, and the product of the midpoint and frequency.

Let's calculate the midpoint for each class interval using the formula: (lower limit + upper limit) / 2.

Class Frequency Midpoint (Midpoint * Frequency)
31-33 18 (31 + 33)/2 = 32 32 * 18 = 576
34-36 25 (34 + 36)/2 = 35 35 * 25 = 875
37-39 22 (37 + 39)/2 = 38 38 * 22 = 836
40-42 9 (40 + 42)/2 = 41 41 * 9 = 369
43-45 20 (43 + 45)/2 = 44 44 * 20 = 880

Now, let's calculate the sum of the frequencies, which is necessary for the mean calculation:
Sum of Frequencies = 18 + 25 + 22 + 9 + 20 = 94

To find the mean, we divide the sum of the products of the midpoint and frequency by the sum of the frequencies:
Mean = ((32 * 18) + (35 * 25) + (38 * 22) + (41 * 9) + (44 * 20)) / 94
= (576 + 875 + 836 + 369 + 880) / 94
= 3536 / 94
= 37.70 (rounded to two decimal places)

To calculate the variance, we need to find the sum of the squared deviations from the mean. Deviation is calculated as the difference between each midpoint and the mean.
Let's calculate the squared deviation for each midpoint:
Squared Deviation = (Midpoint - Mean)^2

Class Frequency Midpoint (Midpoint - Mean)^2
31-33 18 32 (32 - 37.70)^2 ≈ 21.96
34-36 25 35 (35 - 37.70)^2 ≈ 7.29
37-39 22 38 (38 - 37.70)^2 ≈ 0.09
40-42 9 41 (41 - 37.70)^2 ≈ 10.89
43-45 20 44 (44 - 37.70)^2 ≈ 39.69

Now, let's calculate the sum of the squared deviations:
Sum of Squared Deviations = (21.96 * 18) + (7.29 * 25) + (0.09 * 22) + (10.89 * 9) + (39.69 * 20)
≈ 394.08 + 182.25 + 1.98 + 98.01 + 793.80
≈ 1470.12

To find the variance, we divide the sum of the squared deviations by the sum of the frequencies minus 1:
Variance = Sum of Squared Deviations / (Sum of Frequencies - 1)
= 1470.12 / (94 - 1)
= 1470.12 / 93
≈ 15.81 (rounded to two decimal places)

Finally, the standard deviation is the square root of the variance:
Standard Deviation ≈ √15.81
≈ 3.98 (rounded to two decimal places)

Therefore, the mean is approximately 37.70, the variance is approximately 15.81, and the standard deviation is approximately 3.98.