Univariate Data

NZ Level 5

Mean

Lesson

The mean is the **average** of the numbers in a data set.

To calculate the mean, we add up all the scores in a data set, then divide this total by the frequency (ie. number of scores).

Hint

To find the sum of all the scores, we can either add up each individual score, or, if certain scores are repeated, we can add the products of the scores and their frequencies (ie. $f\times x=fx$`f`×`x`=`f``x`). Read through Example 3 to see this in action.

Now let's have a go at calculating the mean of data sets ourselves.

Find the mean of the following scores:

$-14$−14, $0$0, $-2$−2, $-18$−18, $-8$−8, $0$0, $-15$−15, $-1$−1.

Find the mean from the stem-and-leaf plot below:

**Think**: We just need to add up all the scores like before, then divide it by the number of scores.

**Do**:

$\text{Mean }$Mean | $=$= | $\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 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:

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 correct to $2$2 decimal places.

**Think**: We calculate the mean by dividing the sum of the scores (total $fx$`f``x`) by the number of scores (total $f$`f`)

**Do**:

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

$=$= | $48.0851$48.0851... | |

$=$= | $48.09$48.09 |

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`).

Plan and conduct surveys and experiments using the statistical enquiry cycle:– determining appropriate variables and measures;– considering sources of variation;– gathering and cleaning data;– using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets;– comparing sample distributions visually, using measures of centre, spread, and proportion;– presenting a report of findings