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More on Mean, Standard Deviation, Frequency Histogram, Bell Shaped Curve and Empirical Rule

Example:  The following data represent the high-temperature distribution for the month of August in Chicago since 1872.

(a) Approximate the arithmetic mean and standard deviation temperature. 

(b) Draw a frequency histogram of the data to verify that the data are bell shaped.

(c) According to the Empirical Rule, 95% of days in the month of August will be between what two temperatures?

(a.) Press STAT and select 1: Edit. Clear L1 and L2. Enter the midpoints for each of the temperature ranges into L1 and the frequencies (‘days’) into L2. Press STAT highlight CALC, select 1:1-Var Stats, and press 2^nd, 2^nd ENTER.

The population statistics will appear on the screen.

The approximate value of the population mean is 80.43 and the approximate value of the population standard deviation is 8.174.

(b.) To set up the histogram, push 2^nd and ENTER to select Plot 1. Turn ON Plot 1, set Type to Histogram, set Xlist to L1, set Freq to L2.

Press Window to adjust the Graph Window. Set Xmin equal to 54.5 (the midpoint of the first class) and Xmax equal 114.5 (a value that would be the midpoint of an additional class at the end of the table. This extra value is needed to complete the last bar of the histogram). Set Xscl equal to 10, which is the class width. Set Ymin = -5 and Ymax= 1630. You do not need to change Yscl or Xres. Press GRAPH and the histogram should appear.

It may look as if there are only four bars in the histogram. There are actually 6 bars. The first and last bars have such small frequencies compared to the other bars that they are extremely small in the graph. Notice that the histogram is bell shaped.

(c.) The Empirical Rule States that 95% if the data falls in the interval. To calculate the upper and lower limits of this interval, press CLEAR a few times until you get a blank screen. Enter 80.4-2*8.2 to get the lower limit. Press 2^nd ENTER and the calculation will appear again on the screen with the cursor flashing. Move the cursor so that it is positioned on the ‘-‘sign and type in a ‘+’ sign and press ENTER to get the upper limit.