Univariate Data

NZ Level 5

Mean, median, mode and range (combined set)

Lesson

We've already learnt about three measures of central tendency: mean, median and mode. We've also learnt about the range, which is a measure of a data's spread. This chapter is a refresher of all these concepts.

Let's see how much you remember!

The mean is the **average** of all the scores.

You calculate the mean by adding up all the scores, then dividing the total by the number of scores.

Find the mean of the following scores:

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

Think: We need to add up the scores and divide it by the number of scores.

Do:

$\frac{-14+0+\left(-2\right)+\left(-18\right)+\left(-8\right)+0+\left(-15\right)+\left(-1\right)}{8}$−14+0+(−2)+(−18)+(−8)+0+(−15)+(−1)8 | $=$= | $\frac{-58}{8}$−588 |

$=$= | $-7.25$−7.25 |

The median is the **middle score** in a data set.

There are two ways you can find the median:

- Write the numbers in the data set in ascending order, then find the middle score by crossing out a number at each end until you are left with one in the middle.
- Calculate what score would be in the middle using the formula: $\text{middle term }=\frac{n+1}{2}$middle term =
`n`+12, then count up in ascending order until you reach the score that is that term.

Given the following set of scores:

$65.2$65.2, $64.3$64.3, $71.6$71.6, $63.2$63.2, $45.2$45.2, $62.2$62.2, $46.8$46.8, $58.7$58.7

**A)** Sort the scores in ascending order

Think: Ascending means lowest to highest.

Do: $45.2,46.8,58.7,62.2,63.2,64.3,65.2,71.6$45.2,46.8,58.7,62.2,63.2,64.3,65.2,71.6

**B)** Calculate the median, writing your answer as a decimal.

Think: Which term will be in the middle?

Do:

$\text{Middle term }$Middle term | $=$= | $\frac{n+1}{2}$n+12 |

$=$= | $\frac{8+1}{2}$8+12 | |

$=$= | $4.5$4.5 |

This means that the median lies between the fourth and fifth scores.

$\frac{62.2+63.2}{2}$62.2+63.22 | $=$= | $62.7$62.7 |

The median is $62.7$62.7.

The mode is the most frequently occurring score.

To find the mode, just count which score you see most frequently in your data set.

Find the mode of the following set of scores:

$2$2, $2$2, $6$6, $7$7, $7$7, $7$7, $7$7, $11$11, $11$11, $11$11, $13$13, $13$13, $16$16, $16$16

**Think:** How many of each score are there?

**Do: **2, 2, 6, 7, 7, 7, 7, 11, 11, 11, 13, 13, 16, 16

$7$7 is the most frequently occurring score, so the mode is $7$7.

The range is the **difference** between the highest score and the lowest score.

To calculate the range, you need to subtract the lowest score from the highest score.

Find the range of the following set of scores: $10,7,2,14,13,15,11,4$10,7,2,14,13,15,11,4.

Think: What are the highest and lowest scores in this set?

Do: $15-2=13$15−2=13

The range is $13$13.

We can also use the data from graphs to calculate the mean, median, mode and range using the same processes as we have learnt about in the sections above. Let's go through them by looking at an example!

The stem and leaf plot shows the number of hours students spent studying for a science exam.

From the data in the stem and leaf plot, find (to two decimal places if necessary) the:

mean.

median.

mode.

range.

The frequency table below shows the resting heart rate of some people taking part in a study.

Complete the table:

Heart Rate Class Centre ($x$ `x`)Frequency ($f$ `f`)$fx$ `f``x`$30$30-$39$39 $\editable{}$ $13$13 $\editable{}$ $40$40-$49$49 $\editable{}$ $22$22 $\editable{}$ $50$50-$59$59 $\editable{}$ $24$24 $\editable{}$ $60$60-$69$69 $\editable{}$ $36$36 $\editable{}$ Determine an estimate for the mean resting heart rate? Leave your answer to two decimal places if necessary.

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