For the following scores, find the (a) mean, (b) median, (c) sum of squared deviations, (d) variance, and (e) standard deviation:
1,112; 1,245; 1,361; 1,372; 1,472
What don't you understand about this assignment?
To find the mean, median, sum of squared deviations, variance, and standard deviation for the given scores, follow these steps:
Step 1: Calculate the mean (a)
To find the mean, add up all the scores and divide the sum by the total number of scores.
(1,112 + 1,245 + 1,361 + 1,372 + 1,472) / 5 = 5,562 / 5 = 1,112.4
So, the mean (a) is 1,112.4.
Step 2: Calculate the median (b)
The median is the middle value when the scores are arranged in ascending order. If there is an odd number of scores, the median is exactly in the middle. If there is an even number of scores, the median is the average of the two middle values.
Arranging the scores in ascending order: 1,112, 1,245, 1,361, 1,372, 1,472
Since there are five scores, the median is the third value, which is 1,361.
So, the median (b) is 1,361.
Step 3: Calculate the sum of squared deviations (c)
The sum of squared deviations measures the variability of the scores from the mean. To find it, calculate the squared difference between each score and the mean, then sum up all the squared differences.
Squared Deviations = (Score 1 - Mean)^2 + (Score 2 - Mean)^2 + ... + (Score n - Mean)^2
Squared Deviations = (1,112 - 1,112.4)^2 + (1,245 - 1,112.4)^2 + (1,361 - 1,112.4)^2 + (1,372 - 1,112.4)^2 + (1,472 - 1,112.4)^2
Squared Deviations = 0.16 + 16,601.36 + 81,350.24 + 103,470.24 + 132,544.16
Squared Deviations = 333,066.16
So, the sum of squared deviations (c) is 333,066.16.
Step 4: Calculate the variance (d)
The variance is the average of the squared deviations. To find it, divide the sum of squared deviations by the total number of scores.
Variance = Sum of Squared Deviations / Number of Scores
Variance = 333,066.16 / 5
Variance = 66,613.232
So, the variance (d) is 66,613.232.
Step 5: Calculate the standard deviation (e)
The standard deviation is the square root of the variance. To find it, take the square root of the variance.
Standard Deviation = √Variance
Standard Deviation = √66,613.232
Standard Deviation ≈ 257.92
So, the standard deviation (e) is approximately 257.92.