13Aaverages1

= 13A Measures of Central Tendency =


 * Note **
 * In mathematics, the term ** average ** refers to 3 different measures: the mean, median and mode.
 * In the general population, the word average is often used to describe the mean.


 * All three types of average are used to describe where the centre of the data is.


 * The mean is the most commonly used measure.
 * But sometimes the mean is not the best way to describe the centre.
 * A very large ** outlier ** (value a long way from the rest of the data) will distort the mean by a large amount (see example 4 below).
 * A shoe factory might need to know the modal (most common) shoe size, so they make more of that size.

Ungrouped Data

 * Mean **


 * The ** mean ** is calculated by the sum of the values divided by the number of values.

math \\ .\qquad \text{The symbol for mean is } \; \bar{x} \\. \\ . \qquad \qquad \text{ or sometimes, the Greek letter mu: } \mu math

In notation: math \\ . \qquad \bar{x} = \dfrac{\sum{x}}{n} \\. \\ . \qquad \qquad \text {where} \\. \\ . \qquad \qquad \sum{x} = \text{ the sum of the values} \qquad \big\{ \; \Sigma \; \textit{ is the Greek letter Capital Sigma } \big\}. \\ . \qquad \qquad n = \text{ number of values} math


 * Example 1 **

Find the mean of : 3, 4, 6, 8, 9.


 * Solution:**

math . \qquad \sum{x} = 3+4+6+8+9 = 30 \\. \\ . \qquad \text{Number of values } \; n = 5 \\. \\ . \qquad \bar{x} = \dfrac{30}{5} = 6 math


 * Median **


 * The ** median ** is the middle value when the data is in ascending order (from smallest to largest)
 * If there is an even number of values, find the two middle values and take their mean (add them and divide by 2)

math . \qquad \text{The median will be the } \dfrac{n+1}{2} \text{th value when the data is in ascending order} math


 * Example 2a **

Find the median of : 5, 8, 2, 6, 3, 7, 5


 * Solution:**

... ... Put the values in ascending order: 2, 3, 5, 5, 6, 7, 8

... ... Count in from both ends to find the middle value: Median = 5

math . \qquad \text{Note } n = 7, \text{ so median is the } \dfrac{7+1}{2} = 4 \text{th value} \\. \\ . \qquad \text{So count from the left to the 4th value: Median } = 5 math


 * Example 2b **

Find the median of: 5, 1, 8, 9, 3, 8


 * Solution:**

... ... Put the values in ascending order: 1, 3, 5, 8, 8, 9

... ... Count in from both ends to find the middle two numbers are 5, 8

math . \qquad \dfrac{5+8}{2} = 6.5 \qquad \qquad \text{Median } = 6.5 math

math . \qquad \text{Note } n = 6, \text{ so median is the } \dfrac{6+1}{2} = 3.5 \text{th value -- ie between the 3rd and 4th value} \\. \\ . \qquad \text{So count from the left to get the 3rd and 4th values: } 5 \text{ and } 8 \\. \\ . \qquad \dfrac{5+8}{2} = 6.5 \qquad \qquad \text{Median } = 6.5 math


 * Mode **


 * The ** mode ** is the most common value
 * When used as an adjective, the word becomes ** modal **. ... {eg the __modal__ height in the class is ...}
 * A set of data may have:
 * no mode ... ... {no value is more common than any other}
 * one mode
 * two modes ** (bi-modal) ** ... ... {two values are equally the most common}
 * more than two modes ... ... {several values are equally the most common}


 * Example 3 **

Find the mode of: ... ... ** (a) ** .. 1, 3, 5, 7, 9 ... ... ** (b) ** .. 1, 4, 4, 4, 6, 6, 7 ... ... ** (c) ** .. 1, 1, 1, 2, 3, 4, 5, 5, 5


 * Solution:**

... ... ** (a) ** .. There is __**no**__ mode. ... ... ** (b) ** .. Mode = 4 ... ... ** (c) ** .. Mode = 1 and 5 ... ** {Data is bi-modal} **

In a company, the boss earns $311,000 pa. Eleven workers each earn $65,000 and three cleaners earn $48,000 pa. Find ... ... ** (a) ** .. the mean salary ... ... ** (b) ** .. the median salary ... ... ** (c) ** .. the modal salary
 * Example 4 **


 * Solution:**


 * (a) ** .. ** mean salary **

math \\ . \qquad \sum{x} = 311,000 + 11 \times 65,000 + 3 \times 48,000 \\. \\ . \qquad \sum{x} = $1,170,000 \\. \\ . \qquad n = 15 \\. \\ . \qquad \bar{x} = \dfrac{1,170,000}{15} = $78,000 math


 * (b) ** .. ** median salary **

math \\ . \qquad \text{Number of people } n = 15 \\. \\ . \qquad \text{Median is the } \dfrac{15+1}{2} = 8 \text{th person} \\. \\ . \qquad \text{This will be one of the eleven workers, so} \\. \\ . \qquad \text{median salary } = $65,000 math


 * {c} ** .. ** modal salary **

... ... most common salary is that of the 11 workers, so ... ... modal salary = $65,000


 * Note: **
 * Notice how the __**mean**__ is much higher than any of the workers or cleaners earn.
 * This demonstrates how the mean is distorted by a single __**outlier**__ .. (a value a long way from the rest)
 * For situations like this, the __**median salary**__ is usually quoted.

Frequency Tables

 * ** Frequency ** means the number of times each value occurs


 * A ** frequency table ** shows two columns
 * Each possible value ** (x) **
 * The frequency of that value ** (f) **


 * Example 5 **

The ages of young people attending a church youth group is collected into a frequency table: Find ... ... ** (a) ** .. the mean age ... ... ** (b) ** .. the median age ... ... ** (c) ** .. the modal age


 * Solution:**


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


 * To find the mean age, we have to add the ages of all 40 youths in the group.
 * We could do this the long way by adding: 11 + 11 + 11 + 11 + 11 + 11 + 12 + 12 + etc


 * But the quicker way is to recognise that 11 + 11 + 11 + 11 + 11 + 11 = 6 × 11 etc


 * To reduce the possibility of making mistakes, we add an extra column to the frequency table.
 * The extra column is labelled **x** **× f** ... {or age × frequency}
 * Enter the (age × frequency) for each row
 * Then add them up to get the total of all ages.

Hence math . \qquad \text{Mean age is } \bar{x} = \dfrac{528}{40} = 13.2 \text{ years} math


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


 * To find the median age, the values need to be in order.
 * The frequency table automatically puts them in order.

math . \qquad \text{The median person is the } \dfrac{40+1}{2} = 20.5 \text{th person} math


 * This means the median age will be halfway between the ages of the 20th and 21st people.


 * The number aged 11 and under is 6
 * The number aged 12 and under is 6 + 9 = 15
 * The number aged 13 and under is 6 + 9 + 10 = 25


 * Therefore the 20th and 21st person are both 13
 * So, the median age is 13.


 * (c) ** .. ** modal age **


 * The most common age will be the one with the highest frequency
 * So, the modal age is 13

.