calculate the range, variance, standard deviation,
and interquartile range for the data set given {72, 80, 80, 82, 88, 90, 96}
To calculate the range, variance, standard deviation, and interquartile range for the given data set {72, 80, 80, 82, 88, 90, 96}, you can follow these steps:
1. Range:
The range is the difference between the highest and lowest values in the data set.
To calculate the range, subtract the lowest value from the highest value:
Range = 96 - 72 = 24
2. Variance:
The variance measures the average squared distance of each data point from the mean.
To calculate the variance:
Step 1: Find the mean (average):
Add up all the data values and divide the sum by the number of data points:
Mean = (72 + 80 + 80 + 82 + 88 + 90 + 96) / 7 ≈ 84.29
Step 2: Subtract the mean from each data point and square the result:
(72 - 84.29)^2, (80 - 84.29)^2, (80 - 84.29)^2, (82 - 84.29)^2,
(88 - 84.29)^2, (90 - 84.29)^2, (96 - 84.29)^2
Step 3: Find the average of these squared differences:
Add up all the squared differences and divide the sum by the number of data points:
Variance = [(72 - 84.29)^2 + (80 - 84.29)^2 + (80 - 84.29)^2 + (82 - 84.29)^2 + (88 - 84.29)^2 + (90 - 84.29)^2 + (96 - 84.29)^2] / 7 ≈ 52.02
3. Standard Deviation:
The standard deviation is the square root of the variance.
To find the standard deviation, take the square root of the variance:
Standard Deviation = √(52.02) ≈ 7.21
4. Interquartile Range:
The interquartile range represents the range of values in the middle half of the data set.
To calculate the interquartile range:
Step 1: Order the data set in ascending order: 72, 80, 80, 82, 88, 90, 96
Step 2: Find the median (middle value) of the lower half of the data set (Q1) and the upper half (Q3).
In this case, Q1 = 80 and Q3 = 90
Step 3: Find the interquartile range by subtracting Q1 from Q3:
Interquartile Range = Q3 - Q1 = 90 - 80 = 10
So, for the given data set, the calculations are as follows:
Range = 24
Variance ≈ 52.02
Standard Deviation ≈ 7.21
Interquartile Range = 10