Weighted and Grouped on the Graphing Calculator

Weighted Means. Suppose that, in a recent month, a bank customer had $2500 in his account for 5 days, $1900 for 2 days, $1650 for 3 days, $1375 for 9 days, $1200 for 1 day, $900 for 6 days, and $675 for 4 days. To calculate the average balance for that month, you would use a weighted :

∑ ̅ ∑

To do this automatically on a calculator, enter the account balances in L1 and the number of days (weight) in L2:

As before, go to STAT CALC and 1:1-Var Stats. This time, type L1, L2 after 1-Var Stats and ENTER:

The weighted (sample) mean is ̅ (so that the average balance for the month is

$1430.83.) We can also see that the weighted sample is .

Estimating the Mean and Standard Deviation from a Distribution. If your data is organized into a , you can still estimate the mean and standard deviation. For example, suppose that we are given only a frequency distribution of the heights of the 30 males instead of the list of individual heights:

Height (in.) Frequency (f) 62 - 64 3 65- 67 7 68 - 70 9 71 - 73 8 74 - 76 3 ∑ n = 30

We can just calculate the midpoint of each height class and use that midpoint to represent the class. We then find the (weighted) mean and standard deviation for the distribution of midpoints with the given frequencies as in the example above:

Height (in.) Height Class Frequency (f) Midpoint 62 - 64 (62 + 64)/2 = 63 3 65- 67 (65 + 67)/2 = 66 7 68 - 70 69 9 71 - 73 72 8 74 - 76 75 3 ∑ n = 30

The approximate sample mean of the distribution is ̅ , and the approximate sample standard deviation of the distribution is . (These are only estimates of the true mean and standard deviation of the sample because we are basing the calculations on the frequency distribution, which is just a summary of the data, instead of the actual data.)