Question

Cost per serving (in cents) for 15 high-fiber cereals rated very good or good by Consumer Reports are shown below. \(\begin{array}{llllllllllllllll}46 & 49 & 62 & 41 & 19 & 77 & 71 & 30 & 53 & 53 & 67 & 43 & 48 & 28 & 54\end{array}\) Calculate and interpret the mean and standard deviation for this data set.

Short answer

The mean represents the average cost per serving of the cereals. The standard deviation indicates how much the prices deviate from the average price, with a higher value indicating a larger variation in prices.

Step-by-step solution

Calculate the Mean

To calculate the mean, sum up all the elements in the data set, and then divide by the number of elements. Here, the elements are the costs per serving in cents. Use this formula: \[ Mean = \frac{\sum of\:costs}{\# of\:observations} \]

Calculate the Standard Deviation

The standard deviation indicates how much the values within the data set deviate from the mean. The first step to calculate this is to subtract the mean from each cost (in cents), square the result, then sum up all those squared values. Afterwards, divide this sum by the number of observations minus one. Finally, take the square root of this quotient. The formula to use is: \[ Standard\:Deviation = \sqrt{\frac{\sum (cost - mean)^{2}} {\# of\:observations - 1}} \]

Interpret the Results

The interpret the results, remember that the mean represents the average cost per serving, and the standard deviation shows how much the individual costs deviate from this average. A higher standard deviation would indicate a larger spread in the costs per serving.