calculate the range, variance, standard deviation,
and interquartile range for the data set given {72, 80, 80, 82, 88, 90, 96}
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To calculate the range, variance, standard deviation, and interquartile range for the given data set {72, 80, 80, 82, 88, 90, 96}, follow these steps:
1. Range:
The range is the difference between the highest and lowest values in a data set. In this case, the highest value is 96, and the lowest value is 72. So the range can be calculated as follows:
Range = Highest value - Lowest value = 96 - 72 = 24
2. Variance:
Variance measures the spread or dispersion of the data set. To calculate the variance, perform the following steps:
a) Find the mean (average) of the data set. Add up all the values and divide by the number of data points.
Mean = (72 + 80 + 80 + 82 + 88 + 90 + 96) / 7 = 87.14 (rounded to two decimal places).
b) Subtract the mean from each data point and square the result.
For example: (72 - 87.14)^2 = 236.34, (80 - 87.14)^2 = 51.46, and so on.
c) Add up all the squared differences.
Sum of squared differences = 236.34 + 51.46 + 51.46 + 24.58 + 0.02 + 5.46 + 64.42 = 433.74 (rounded to two decimal places).
d) Divide the sum of squared differences by the number of data points (n).
Variance = Sum of squared differences / n = 433.74 / 7 = 61.96 (rounded to two decimal places).
3. Standard Deviation:
The standard deviation is the square root of the variance. To calculate the standard deviation, take the square root of the previously calculated variance.
Standard Deviation = √Variance = √61.96 = 7.87 (rounded to two decimal places).
4. Interquartile Range:
The interquartile range measures the spread of the middle 50% of the data. To calculate the interquartile range, perform the following steps:
a) Arrange the data set in ascending order: 72, 80, 80, 82, 88, 90, 96.
b) Find the median, which is the middle value. In this case, there are an odd number of data points, so the median is the value in the middle position. The median is 82.
c) Split the data set into two halves, excluding the median.
Lower half: 72, 80, 80.
Upper half: 88, 90, 96.
d) Find the median of each half.
Lower quartile (Q1): Median of the lower half = 80.
Upper quartile (Q3): Median of the upper half = 90.
e) Subtract the lower quartile from the upper quartile.
Interquartile Range = Q3 - Q1 = 90 - 80 = 10.
Therefore, the calculated values for the given data set are:
Range = 24
Variance = 61.96
Standard Deviation = 7.87
Interquartile Range = 10.