13Aaverages2

= 13A Measures of Centre =

Grouped Data

 * When there is a large range of possible values
 * or the data is continuous (resulting from measurement)
 * the data is often grouped into classes within a frequency table


 * Each class (or group) should have the same class interval (upper limit minus lower limit)


 * Notation **

... ... 10 --< 20 ... means the value is somewhere from 10 up to, but not including 20. math . \qquad \qquad \qquad \text{ or } \;\; 10 \leq x < 20 math


 * Note **
 * When data is collected into groups (or classes), we lose information about the exact results.
 * Therefore we can only find approximate measures of the centre.
 * We use the class centre (middle of the group) to calculate those approximate measures


 * Example 1 **

The ages of people attending a community festival were collected into a table with a class interval of 10. Find ... ... ** (a) ** .. the approximate mean age ... ... ** (b) ** .. the approximate median age ... ... ** (c) ** .. the modal class


 * Solution:**


 * (a) ** .. ** approximate mean age **

... ... add a new column to the table showing the middle of each class math . \qquad \text{for a class given by } \; \text{ a --< b } \; \text{ the middle can be found by } \dfrac{a+b}{2} \\. \\ . \qquad \text{so for the class, 10 --< 20, the class centre is } \dfrac{10+20}{2} = 15 math

... ... then add a new column for .. class middle × frequency ... ** (x ** mid ** × f) **



math .\qquad \text{so, the approximate mean is } \bar{x} = \dfrac{7250}{400} \approx 18.1 math

... ... Notice that this is only approximate because it is calculated from the class centres rather than the original data


 * (b) ** .. ** approximate median age **

math \\ . \qquad \text{the number of values is } \; n = 400 \\. \\ . \qquad \text{so the median value is the } \dfrac{400+1}{2} = 200.5 \text{th person} \\. \\ math

... ... there are 150 people under 10 years old. ... ... there are 150 + 70 = 220 people under 20 years old.

... ... so, the 200.5 th person is in the class 10 --< 20 ... ... take the middle value of the class

... ... **so the median age is approximately 15**

If you want to make your estimate for the median a little more accurate, you can do this:
 * Note **
 * the 200.5 th person is the 200.5 – 150 = 50.5 th person in the class out of 70

math . \qquad \text{or } \; \dfrac{50.5}{70} \approx 0.72 \text{ of the way through the group} math


 * the class interval is 10 (upper limit minus lower limit)
 * so, the median is 0.72 x 10 = 7.2 through the group


 * **so, a better approximate median age is 17.2**

... ... {the textbook does this by drawing a graph and reading an approximate value from the graph}.


 * (b) ** .. ** modal class **

... ... the modal class is the class with the highest frequency

... ... **the modal class is 0 --< 10** .