The following data lists the average monthly snowfall for January in 15 cities around the US:
37 33 35 27 9 45 22 8
42 48 37 39 9 15 12
Find the mean, variance, and standard deviation
Find the mean first = sum of scores/number of 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.
To find the mean, variance, and standard deviation of the given data, follow these steps:
Step 1: Find the total sum of the data values.
Add up all the values in the data set:
37 + 33 + 35 + 27 + 9 + 45 + 22 + 8 + 42 + 48 + 37 + 39 + 9 + 15 + 12 = 438
Step 2: Find the mean (average) of the data set.
Divide the sum by the number of values in the data set.
Mean = Sum / Number of values
Mean = 438 / 15 = 29.2 (rounded to one decimal place)
So, the mean snowfall for January in the 15 cities is 29.2 inches.
Step 3: Find the variance.
Variance measures the spread or dispersion of the data set.
The formula for variance is:
Variance = (sum of squared differences from the mean) / (number of values)
First, find the squared differences from the mean for each value:
(37 - 29.2)^2 = 54.76
(33 - 29.2)^2 = 14.44
(35 - 29.2)^2 = 33.64
(27 - 29.2)^2 = 4.84
(9 - 29.2)^2 = 416.16
(45 - 29.2)^2 = 251.84
(22 - 29.2)^2 = 52.36
(8 - 29.2)^2 = 441.6
(42 - 29.2)^2 = 163.84
(48 - 29.2)^2 = 352.84
(37 - 29.2)^2 = 54.76
(39 - 29.2)^2 = 96.04
(9 - 29.2)^2 = 416.16
(15 - 29.2)^2 = 201.64
(12 - 29.2)^2 = 295.84
Now, calculate the sum of these squared differences:
54.76 + 14.44 + 33.64 + 4.84 + 416.16 + 251.84 + 52.36 + 441.6 + 163.84 + 352.84 + 54.76 + 96.04 + 416.16 + 201.64 + 295.84 = 3,279.52
Next, divide the sum by the number of values:
Variance = 3,279.52 / 15 = 218.64 (rounded to two decimal places)
So, the variance of the snowfall for January in the 15 cities is 218.64 square inches.
Step 4: Find the standard deviation.
The standard deviation is the square root of the variance and provides a measure of how spread out the values are from the mean.
Standard Deviation = √Variance
Standard Deviation = √218.64 ≈ 14.79 (rounded to two decimal places)
Therefore, the standard deviation of the snowfall for January in the 15 cities is approximately 14.79 inches.