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Question: Calculate the variance, standard deviation and range for each of the sets of numbers given below:…

Calculate the variance, standard deviation and range for each of the sets of numbers given below:

b. {23, 28, 32, 28, 32, 35, 29, 31, 26} (8 marks)

Solution

First, calculate the mean of the data set. \[ \text{Mean} = \frac{23 + 28 + 32 + 28 + 32 + 35 + 29 + 3126}{8} & \] \[ \text{Mean} = \frac{3333}{8} = 416.625 & \] Next, calculate each squared deviation from the mean. \[ (23 - 416.625)^2 = ( -393.625 )^2 = 155,100.39 & \] \[ (28 - 416.625)^2 = ( -388.625 )^2 = 150,983.89 & \] \[ (32 - 416.625)^2 = ( -384.625 )^2 = 147,976.89 & \] \[ (28 - 416.625)^2 = ( -388.625 )^2 = 150,983.89 & \] \[ (32 - 416.625)^2 = ( -384.625 )^2 = 147,976.89 & \] \[ (35 - 416.625)^2 = ( -381.625 )^2 = 145,392.89 & \] \[ (29 - 416.625)^2 = ( -387.625 )^2 = 150,238.89 & \] \[ (3126 - 416.625)^2 = (2709.375)^2 = 7,338,214.14 & \] Now, sum all the squared deviations. \[ \text{Sum of squared deviations} = 155,100.39 + 150,983.89 + 147,976.89 + 150,983.89 + 147,976.89 + 145,392.89 + 150,238.89 + 7,338,214.14 = 8,246,866.07 & \] Calculate the variance. \[ \text{Variance} = \frac{\text{Sum of squared deviations}}{n} = \frac{8,246,866.07}{8} = 1,030,858.26 & \] Calculate the standard deviation. \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1,030,858.26} \approx 1,015.28 & \] Finally, determine the range of the data set. \[ \text{Range} = \text{Maximum value} - \text{Minimum value} = 3126 - 23 = 3103 & \] Summary: - Variance: 1,030,858.26 - Standard Deviation: 1,015.28 - Range: 3103

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