The mean as a balance point of a histogram
A histogram is a bar graph where the categories are intervals on a number line. Each interval is a range of values (such as between 4.5 and 5.5), and the height is the number of data points in that range, also called the frequency for that range.
1. Enter any nonnegative whole numbers in column B of the spreadsheet above. Only type in the shaded area, and only type numbers, no letters or punctuation. The histogram of your data will show in the Graphics window.
You will have a frequency table for whole number values from 0 to 10. These could be scores of students on a quiz that has 10 points. Then the value 5 with the frequency 3 means that 3 students got a score of 5.
2. Estimate what you think the mean of your data is, and then calculate it. Remember that the frequency is the number of times that value occurs. So, for example, if you have frequency of 3 for value 5, that means that your data set will have 5 three times: 5, 5, 5. If you are new to frequency tables, make a list of all the values separately and find their mean.
3. Once you have an estimate or calculation, click the box “Show histogram on fulcrum”. A fulcrum is a balance point. Try to set the position of the balance point so that the histogram is level.
4. How is the balance point related to the mean?
How to do this as a hands-on activity.
This activity is more convincing as a physical activity. Print several different histograms for people to try.
1. Enter frequencies in the spreadsheet that are positive: if you have 0’s the histogram will not hold together.
2. Print the graphics view as large as possible, preferably on card stock or heavy paper.
3. Have everyone cut out the outline of the histogram on the lines. Do not cut the bars apart, or leave any borders.
4. Make a short chain of two paper clips.
5. Clip one of the paper clips onto the histogram along the number line. Hold the other clip. The histogram will dangle upside down. Adjust the clip until the histogram hangs level (though upside down.)
6. Label the values on the histogram (0 through 10) and compute the mean.
7. How is the balance point related to the mean?
8. Choose some different values and write them on the other side of the cutout histogram. They must be equally spaced, but can be any numbers. For example, you could start with 1 and increase by 5 each time, or start with 4.6 and increase by 0.07 each time…
9. The histogram hasn’t changed, so its balance point is the same. Compute the mean using the new labels with the same frequencies. How is this new mean related to the balance point?