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?

Considering the total number of points, what was the mean number of points scored each game? Round to 2 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 (call it $x$`x`).

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