Averages from Grouped Frequency Tables

A grouped frequency table puts the data into groups called class intervals, or classes. It shows how many values are in each class, but not the values themselves.

Here 18 people recorded how long they spent reading one evening. 10<t\le20 means more than 10 minutes, up to and including 20 minutes. So exactly 20 minutes goes in this class, but exactly 10 minutes goes in 0<t\le10

Because the exact values are lost, you cannot find the exact mean, median or range. You can find the modal class, the class that contains the median, and an estimate of the mean.

In the exam you are usually asked to write down the modal class, or to find the class interval that contains the median, for 1 mark each, and to work out an estimate for the mean, for 3 marks. Harder questions ask why the answer is only an estimate, ask you to use the estimate to make a decision, or ask whether someone's statement about the median is right.

The modal class is the class with the highest frequency. Write down the class itself, such as 10<t\le20, not its frequency.

Add up the frequencies to find how many people or things there are. This is the total frequency. You need it for the median and for the mean.

The median is the middle value when the data is in order. A grouped table doesn't show the exact values, so you can only find which class the median is in. So don't give the middle of a class, such as 25 for 20<t\le30, as the median.

If there are n values, the median is the value in position

\frac{n+1}{2}

Keep a running total of the frequencies as you go down the table. The median is in the first class whose running total reaches that position. For 60 values the position is 30.5, so look for the class that holds the 30th and 31st values.

The class that contains the median is often not the middle row of the table. Count the people, not the rows. In the exam this is asked as “Find the class interval that contains the median”.

A grouped table doesn't give the exact values, so you use the midpoint of each class to stand for every value in it. The midpoint is halfway between the two ends of the class:

\text{midpoint}=\frac{\text{lower end}+\text{upper end}}{2}

For 10<t\le20 the midpoint is (10+20)\div2=15. For 5<t\le10 it is (5+10)\div2=7.5. For whole-number classes, such as 5 to 9 books, it is (5+9)\div2=7

Midpoint × frequency gives an estimate of the total for one class. In the reading table, 9 people read for more than 10 and up to 20 minutes. Between them they read for about 15\times9=135 minutes.

Add up the midpoint × frequency column. This is an estimate of the total of all the data. Then divide by the total frequency:

\text{estimate of the mean}=\frac{\text{total of midpoint}\times\text{frequency}}{\text{total frequency}}

Divide by the total frequency, the number of people. Don't divide by the number of classes, and don't divide by the total of the midpoints.

Keep every product exact and round only the final answer. Then check it: the estimate must lie between the bottom of the first class and the top of the last class, and it is usually near the classes with the biggest frequencies.

If a few values are much bigger than the rest, they pull the mean up. Then the modal class or the class that contains the median may describe a typical value better than the mean does.

In the reading table, the 5 people in 0<t\le10 could all have read for 1 minute, or all for 10 minutes. Using the midpoint assumes that, on average, they read for 5 minutes.

So a mean worked out from midpoints is only an estimate. If a question asks why, say that the exact values are not known, so the midpoint of each class was used instead.

The range can only be estimated too. The largest it could be is the top of the last class minus the bottom of the first class, using only classes with a frequency above 0. For the reading table that is 30-0=30 minutes.

• Dividing by the number of classes. Divide the total of midpoint × frequency by the total frequency.
• Adding the midpoints or the frequencies and dividing. Multiply each midpoint by its frequency first, then add the products.
• Using the end of each class instead of the midpoint. Use the value halfway along each class, such as 15 for 10<t\le20
• Taking the middle row of the table as the median class. Find the position (n+1)\div2, then use a running total of the frequencies.
• Going one class too far for the median. Stop at the first class whose running total reaches the position.
• Rounding before the end. Keep the frequencies and every product exact, and round only the final answer.