the following data lists the average monthly snowfall for jan 15 cities around us:
34 9 28 24 0 46 34 20
1 38 5 16 32 26 25
find the mean, variance, and standard deviation
Mean = 22.53
Variance = 13.8969
Standard deviation = 3.72785
To find the mean, variance, and standard deviation, we can follow these steps:
Step 1: Find the mean:
The mean is the average of a set of numbers. To find the mean, we need to sum up all the numbers in the data set and then divide the sum by the total number of values.
The data set for the average monthly snowfall is:
34, 9, 28, 24, 0, 46, 34, 20, 1, 38, 5, 16, 32, 26, 25
Adding up all the numbers, we get: 34 + 9 + 28 + 24 + 0 + 46 + 34 + 20 + 1 + 38 + 5 + 16 + 32 + 26 + 25 = 378
The total number of values is 15.
Therefore, the mean snowfall is: 378 / 15 = 25.2 inches
Step 2: Find the variance:
The variance measures how spread out the values in a data set are from the mean. To find the variance, we need to calculate the squared difference between each value and the mean, sum up all the squared differences, and then divide by the total number of values.
First, calculate the squared difference for each value:
(34 - 25.2)^2, (9 - 25.2)^2, (28 - 25.2)^2, (24 - 25.2)^2, (0 - 25.2)^2, (46 - 25.2)^2, (34 - 25.2)^2, (20 - 25.2)^2, (1 - 25.2)^2, (38 - 25.2)^2, (5 - 25.2)^2, (16 - 25.2)^2, (32 - 25.2)^2, (26 - 25.2)^2, (25 - 25.2)^2
Calculating the squared differences, we get:
81.44, 278.89, 3.24, 1.44, 635.04, 388.09, 81.44, 24.01, 577.44, 156.25, 395.84, 78.44, 53.44, 0.04, 4.84
Summing up all the squared differences, we get: 81.44 + 278.89 + 3.24 + 1.44 + 635.04 + 388.09 + 81.44 + 24.01 + 577.44 + 156.25 + 395.84 + 78.44 + 53.44 + 0.04 + 4.84 = 2749.94
The total number of values is 15.
Therefore, the variance is: 2749.94 / 15 = 183.33 square inches (rounded to two decimal places)
Step 3: Find the standard deviation:
The standard deviation is the square root of the variance. To find the standard deviation, we can take the square root of the variance calculated in step 2.
Taking the square root of the variance, we get: √183.33 = 13.54 inches (rounded to two decimal places)
So, the mean snowfall is 25.2 inches, the variance is 183.33 square inches, and the standard deviation is 13.54 inches.