 New Zealand
Level 6 - NCEA Level 1

5 Number Summary

Lesson

In Quarter & Spread, we looked at quartiles and found that when we quarter a data set, we can determine 5 key values. These are the minimum value, the lower quartile, the median, the upper quartile and the maximum value. So here is the 5 number summary:

Minimum value

The minimum value is the lowest score in a data set.

Lower quartile

The lower quartile is also called the first quartile. It is the middle score between the lowest score and the median and it represents the $25$25th percentile.

The lower quartile is the $\frac{n+1}{4}$n+14th score, where $n$n is the total number of scores.

Median

The median is the middle score in a data set.

It is calculated as the $\frac{n+1}{2}$n+12th score, where $n$n is the total number of scores.

Upper quartile

The upper quartile is also called the third quartile. It is the middle score between the highest score and the median and it represents the $75$75th percentile.

The upper quartile is the $\frac{3\left(n+1\right)}{4}$3(n+1)4th score, where $n$n is the total number of scores.

Maximum value

The maximum value is the highest score in a data set.

What it all means

The five numbers from the five number summary break up our set of scores into four parts. Have a look at the diagram here: So knowing the five numbers can help us identify key regions of $25%$25%, $50%$50%, and $75%$75% of the scores.

It also leads nicely to the development of box plots that we will learn about soon.

Practice questions

Question 1`

The table shows the number of points scored by a basketball team in each game of their previous season.

 $59$59 $67$67 $73$73 $82$82 $91$91 $58$58 $79$79 $88$88 $69$69 $84$84 $55$55 $80$80 $98$98 $64$64 $82$82
1. Sort the data in ascending order.

2. State the maximum value of the set.

3. State the minimum value of the set.

4. Find the median value.

5. Find the lower quartile.

6. Find the upper quartile.

Question 2

To gain a place in the main race of a car rally, teams must compete in a qualifying round. The median time in the qualifying round determines the cut off time to make it through to the main race. Below are some results from the qualifying round.

$75%$75% of teams finished in $159$159 minutes or less.

$25%$25% of teams finished in $132$132 minutes or less.

$25%$25% of teams finished between with a time between $132$132 and $142$142 minutes.

1. Determine the cut off time required in the first round to make it through to the main race.

2. Determine the interquartile range in the qualifying round.

3. In the qualifying round, the ground was wet, while in the main race, the ground was dry. To make the times more comparable, the finishing time of each team from the qualifying round is reduced by $5$5 minutes. What would be the new median time from the qualifying round?

Outcomes

S6-1

Plan and conduct investigations using the statistical enquiry cycle: A justifying the variables and measures used B managing sources of variation, including through the use of random sampling C identifying and communicating features in context (trends, relationships between variables, and differences within and between distributions), using multiple displays D making informal inferences about populations from sample data E justifying findings, using displays and measures.

91035

Investigate a given multivariate data set using the statistical enquiry cycle