Question

Estimating Summary Statistics For the dataset $$ 45,46,48,49,49,50,50,52,52,54,57,57,58,58,60,61 $$ (a) Without doing any calculations, estimate which of the following numbers is closest to the mean: $$ 60,53,47,58 $$ (b) Without doing any calculations, estimate which of the following numbers is closest to the standard deviation: $$ \begin{array}{lllll} 52, & 5, & 1, & 10, & 55 \end{array} $$ (c) Use technology to find the mean and the standard deviation for this dataset.

Short answer

The estimated mean of the dataset is 53 and the estimated standard deviation is 10. The calculated mean is 53 and the standard deviation needs to be calculated using the formula which involves subtracting each value by mean, squaring each result, obtaining an average of these squared values and finally taking the square root.

Step-by-step solution

Estimate Mean

In the dataset, small numbers and high numbers balance around the middle. Therefore, pick the middle value from options 60, 53, 47, 58. Hence, 53 is closest to the mean.

Estimate Standard Deviation

Standard deviation is the spread of data from the mean. Therefore, observing the dataset, we can see the spread is more than 1 but less than 52. Hence, among the options 52, 5, 1, 10, 55, 10 is closest to the standard deviation.

Calculate Mean

Add all the given numbers of the dataset and divide by the number count. That is, \(\frac{45 + 46 + 48 + 49 + 49 + 50 + 50 + 52 + 52 + 54 + 57 + 57 + 58 + 58 + 60 + 61 }{16}\). The average comes out to be around 53.

Calculate Standard deviation

Subtract each data point from the mean, square it, add those, divide by the number total minus 1, and then take the square root, to obtain the standard deviation.