9. Centre and Spread

Lesson

The mean is described as the **average** of the numbers in a data set. It is defined as the sum of the scores divided by the number of scores.

The symbol for the mean of a sample is $\overline{x}$`x`, whilst the population mean is represented by the symbol $\mu$`μ` (Greek letter 'mu'). We typically don't have data for every member of the population, so we usually don't know $\mu$`μ` exactly, but we can estimate it by using the sample mean, $\overline{x}$`x`, from a well designed survey.

If certain scores are repeated, such as when information is given in a frequency table then we can find the total sum of all scores by multiplying each **unique** score by its frequency, then adding them all up.

We summarise the calculation of the mean below.

Mean

The mean of a set of data is calculated by:

$\text{Mean}=\frac{\text{Total sum of all scores}}{\text{Number of scores}}$Mean=Total sum of all scoresNumber of scores

If certain scores are repeated, then:

$\text{Total sum of all scores}=\text{sum of}\ \left(\text{Unique score}\times\text{Frequency}\right)$Total sum of all scores=sum of (Unique score×Frequency)

Now let's look at a few examples of calculating the mean of different data sets.

Find the mean from the data in the stem plot below.

Stem | Leaf | |||

$2$2 | $3$3 | $8$8 | ||

$3$3 | $1$1 | $1$1 | $1$1 | |

$4$4 | $0$0 | $3$3 | ||

$5$5 | $0$0 | $3$3 | $8$8 | $8$8 |

$6$6 | $2$2 | $2$2 | $9$9 | |

$7$7 | $1$1 | $8$8 | ||

$8$8 | $3$3 | |||

$9$9 | $0$0 | $0$0 | $1$1 |

**Think:** We can find the mean by adding up all of the scores, then dividing the total by the number of scores.

**Do:**

$\text{Mean}$Mean | $=$= | $\frac{\text{Total of all scores}}{\text{Number of scores}}$Total of all scoresNumber of scores |

$=$= | $\frac{23+28+3\times31+40+43+50+53+2\times58+2\times62+69+71+78+83+2\times90+91}{20}$23+28+3×31+40+43+50+53+2×58+2×62+69+71+78+83+2×90+9120 | |

$=$= | $\frac{1142}{20}$114220 | |

$=$= | $57.1$57.1 |

A statistician has organised a set of data into the frequency table shown.

Score ($x$x) |
Frequency ($f$f) |
---|---|

$44$44 | $8$8 |

$46$46 | $10$10 |

$48$48 | $6$6 |

$50$50 | $18$18 |

$52$52 | $5$5 |

**(a)** Complete the frequency distribution table by adding a column showing the total sum for each unique score.

**Think:** For each unique score ($x$`x`-value), multiply it by the number of times that score appears. In other words, multiply the unique score by its frequency $\left(f\right)$(`f`) to find the total sum for that score.

**Do:** So for a score of $44$44, which occurred $8$8 times, the total score is $44\times8=352$44×8=352. Completing the entire column, we get the following table.

Score ($x$x) |
Frequency ($f$f) |
$fx$fx |
---|---|---|

$44$44 | $8$8 | $352$352 |

$46$46 | $10$10 | $460$460 |

$48$48 | $6$6 | $288$288 |

$50$50 | $18$18 | $900$900 |

$52$52 | $5$5 | $260$260 |

Totals |
$47$47 | $2260$2260 |

**(b) **Calculate the mean of this data set. Round your answer to two decimal places.

**Think:** We calculate the mean by dividing the sum of the scores (that is, the sum of all the $fx$`f``x`'s) by the number of scores (the total frequency).

**Do:**

$\text{Mean}$Mean | $=$= | $\frac{\text{Total of all scores}}{\text{Number of scores}}$Total of all scoresNumber of scores |

$=$= | $\frac{2260}{47}$226047 | |

$=$= | $48.09$48.09 ($2$2 d.p.) |

Using technology

Throughout this chapter and in particular for moderate to large data sets, you should use appropriate technology such as a calculator with a statistics program or your computer.

Tips:

- Familiarise yourself with the program and the types of calculations and graphs it is capable of creating.
- Ensure settings are correct for the data given, this is particularly important when changing between data that is in a simple list to data that is in a frequency table.
- Take note of the different symbols used for the different calculations we will encounter.

Find the mean of the following scores:

$8$8, $15$15, $6$6, $27$27, $3$3.

In each game of the season, a basketball team recorded the number of 'three-point shots' they scored. The results for the season are represented in the given dot plot.

What was the total number of points scored from three-point shots during the season?

What was the mean number of points scored from three-point shots each game? Round to two decimal places if necessary.

What was the mean number of three point shots per game this season? Leave your answer to two decimal places if necessary.

The mean of $4$4 scores is $21$21. If three of the scores are $17$17, $3$3 and $8$8, find the $4$4th score, $x$`x`.

Enter each line of working as an equation.

calculate measures of central tendency, the arithmetic mean and the median