18.03 Summarising data

Exploration

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.

Worked example

Exploration

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.

Exploration

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.

Exploration

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:

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:

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

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.