Answer:Therefore, the range is 32 and the standard deviation is approximately 10.523 for the given data set.
Step-by-step explanation:To calculate the range and standard deviation for the given data set {40, 65, 33, 46, 55, 50, 61}, we can follow these steps:
Step 1: Find the range.
The range is calculated by subtracting the minimum value from the maximum value in the data set.
Range = Maximum value - Minimum value
Range = 65 - 33
Range = 32
Step 2: Calculate the mean.
The mean is calculated by summing up all the values in the data set and dividing by the total number of values.
Mean = (40 + 65 + 33 + 46 + 55 + 50 + 61) / 7
Mean = 350 / 7
Mean = 50
Step 3: Calculate the deviation for each value.
Deviation is calculated by subtracting the mean from each value in the data set.
Deviation for 40 = 40 - 50 = -10
Deviation for 65 = 65 - 50 = 15
Deviation for 33 = 33 - 50 = -17
Deviation for 46 = 46 - 50 = -4
Deviation for 55 = 55 - 50 = 5
Deviation for 50 = 50 - 50 = 0
Deviation for 61 = 61 - 50 = 11
Step 4: Square each deviation.
Squared deviation for -10 = (-10)^2 = 100
Squared deviation for 15 = 15^2 = 225
Squared deviation for -17 = (-17)^2 = 289
Squared deviation for -4 = (-4)^2 = 16
Squared deviation for 5 = 5^2 = 25
Squared deviation for 0 = 0^2 = 0
Squared deviation for 11 = 11^2 = 121
Step 5: Calculate the variance.
Variance is calculated by summing up all the squared deviations and dividing by the total number of values.
Variance = (100 + 225 + 289 + 16 + 25 + 0 + 121) / 7
Variance = 776 / 7
Variance ≈ 110.857
Step 6: Calculate the standard deviation.
The standard deviation is the square root of the variance.
Standard Deviation ≈ √110.857
Standard Deviation ≈ 10.523