Lesson

Statistical data can be divided two types: categorical and numerical. There are four ways of summarising numerical data: the mode, mean, median and range.

Data that is collected as a set of words is called categorical data.

Imagine asking someone for their favourite colour, country of birth, or gender. Their answer would always be a word. We can also think of categorical data as values which can be sorted into groups or categories.

When the data is a set of numbers, it is called numerical data.

Imagine asking someone for their height, their age, or how long they spend on social media each day. Their answers would always be a number.

Numerical data is divided into two types: continuous and discrete.

Discrete numerical data is counted, so its values are separated. If you asked someone to tell you how many pets they have they might say "$4$4", but they would not say "$4$4 and seven sixteenths".

Continuous data is measured, so it can take any value within a range - there are an infinite number of possible values. If we measure an animal's height, we might find any reasonable value, limited only by the precision of our ruler.

Data types

Categorical data is made up of words.

Numerical data is made up of numbers.

- Discrete numerical data is counted.
- Continuous numerical data is measured.

The mode of a data set is the most commonly occurring score.

A class quiz is marked out of $10$10, and ten students receive the following marks:

$10,7,6,9,7,8,6,7,7,8$10,7,6,9,7,8,6,7,7,8

To find the mode we can count how many times each score occurred (the frequency). It can help to order the scores first:

$6,6,7,7,7,7,8,8,9,10$6,6,7,7,7,7,8,8,9,10

The score $6$6 appears twice, which means it has a frequency of $2$2. Similarly, $7$7 has a frequency of $4$4, $8$8 has a frequency of $2$2 and $9$9 and $10$10 each have a frequency of $1$1.

This means that $7$7 has the highest frequency, therefore the mode is $7$7.

Mode

The mode of a data set is the result with the highest frequency.

The frequency is the number of times that a score occurs.

If there are multiple results that share the highest frequency then there will be more than one mode.

The mean of a data set is an average score.

Three friends are planning a trip to Alice Springs. They plan to fly there, and discover that the airline imposes a weight limit on their luggage of $20$20 kg per person. On the night before the flight they weigh their luggage and find that their luggage weights form this data set:

$17,18,22$17,18,22

One of them has packed too much. They decide to share their luggage around so that they all carry the same amount. How much does each person carry now?

Thinking about it using more mathematical language, we are sharing the total luggage equally among three groups. As a mathematical expression, we find:

$\frac{17+18+22}{3}=\frac{57}{3}=19$17+18+223=573=19

Each person carries $19$19 kg. This amount is the **mean** of the data set.

Summary

If we replace every number in a numerical data set with the mean, the sum of the numbers in the data set will be the same.

To calculate the mean, use the formula:

$\text{mean}=\frac{\text{sum of scores}}{\text{number of scores}}$mean=sum of scoresnumber of scores

Find the mean of this data set:

$4,7,1,2,3$4,7,1,2,3

**Think: **There are $5$5 scores, so we should add these numbers all together and divide by $5$5.

**Do: **$4+7+1+2+3=17$4+7+1+2+3=17, and $17\div5=3.4$17÷5=3.4.

**Reflect: **Even though all the numbers in the data set are whole numbers, the mean is a decimal. If the data set was produced from a survey question "How many siblings do you have?", we would say the mean number of siblings was $3.4$3.4, even though it isn't possible to have $0.4$0.4 siblings! The mean is a way to summarise data - it is not part of the data set itself.

The median of a data set is another kind of average.

Seven people were asked about their weekly income, and their responses form this data set:

$\$300,\$400,\$400,\$430,\$470,\$490,\$2900$$300,$400,$400,$430,$470,$490,$2900

The mean of this data set is $\frac{\$5390}{7}=\$770$$53907=$770, but this amount doesn't represent the data set very well. Six out of seven people earn much less than this.

Instead we can select the **median**, which is the middle score. We remove the biggest and the smallest scores:

$\$400,\$400,\$430,\$470,\$490$$400,$400,$430,$470,$490

Then the next biggest and the next smallest:

$\$400,\$430,\$470$$400,$430,$470

Then the next biggest and the next smallest:

$\$430$$430

There is only one number left, and this is the **median** - so for this data set the median is $\$430$$430. This weekly income is much closer to the other scores in the data set, and summarises the set better.

It helped that this set was already in order, and that there were an odd number of scores. What happens when this isn't the case?

Six people were asked to count the number of advertisements they saw while browsing the internet for an hour, and their responses form this data set:

$96,39,0,40,33,27$96,39,0,40,33,27

To find the median let's do the same thing we did before - we remove the biggest and the smallest scores:

$39,40,33,27$39,40,33,27

And the next biggest and the next smallest:

$39,33$39,33

Now that we are down to two scores, we find the number directly in between them. We can add the numbers together and divide by $2$2, just like finding a mean:

$\frac{39+33}{2}=\frac{72}{2}=36$39+332=722=36

The median number of advertisements that the six people saw was $36$36. This means that $50%$50% of people saw more than $36$36, and $50%$50% saw less than $36$36.

Finding the median of an ordered data set is much easier - if the set is scrambled up, you may want to rewrite it in order first.

Summary

The median of a numerical data set is the "middle" score, and its definition changes depending on the number of scores in the data set.

If there are an odd number of scores, the median will be the middle score.

If there are an even number of scores, the median will be the number in between the middle two scores, and half the scores will be greater than the median, and half will be less than the median.

The range is the difference between the highest and the lowest score in a data set. Unlike the **mean** and the **median**, the range doesn't measure the center - instead it measures how spread out it is.

Two bus drivers, Kenji and Björn, track how many passengers board their buses each day for a week. Their results are displayed in this table:

M | T | W | T | F | |
---|---|---|---|---|---|

Kenji | $10$10 | $13$13 | $14$14 | $16$16 | $11$11 |

Björn | $2$2 | $27$27 | $13$13 | $5$5 | $17$17 |

Both data sets have the same median and the same mean, but the sets are quite different. To calculate the range, we start by finding the highest and lowest number of passengers for each driver:

Highest | Lowest | |
---|---|---|

Kenji | $16$16 | $10$10 |

Björn | $27$27 | $2$2 |

Now we subtract the lowest from the highest to find the difference, which is the **range**:

Range | |||
---|---|---|---|

Kenji | $16-10$16−10 | $=$= | $6$6 |

Björn | $27-2$27−2 | $=$= | $25$25 |

Notice how Kenji's range is quite small, at least compared to Björn's. We might say that Kenji's route is more predictable, and that Björn's route is much more variable.

We can see that the range does not say anything about the size of the scores, just their spread.

Summary

The range of a numerical data set is the difference between the highest and the lowest score.

$\text{Range}=\text{Highest score}-\text{Lowest score}$Range=Highest score−Lowest score

Find the mean of the following scores:

$4,8,2,5,1$4,8,2,5,1

Find the median of the following scores:

$3.2,2.3,5.5,4.6,8.5$3.2,2.3,5.5,4.6,8.5

Find the range of the following scores:

$11,-19,14,17,-11,15,13,-5,-20$11,−19,14,17,−11,15,13,−5,−20

collects, represents and interprets single sets of data, using appropriate statistical displays

analyses single sets of data using measures of location, and range