Mean, median, mode and range

Mean, median, mode and range.

Average screen time. Average wage. A striker's average goals per game. You get thrown that word every single day. And average can mean three completely different numbers, and whoever writes the headline gets to pick which one. Nine people work in a shop. Eight of them earn twenty thousand pounds a year. The owner pays himself two hundred thousand. So the average wage in that shop is exactly forty thousand pounds. That is true. It is also true that almost everyone there earns twenty thousand. Same nine numbers. Two honest answers. Two completely different stories.

So which average did they use? Mean, median, mode and range. Four words, one list of numbers, four different answers. Let's take one set of data and get all four out of it.

This is the first video in our chapter on averages, spread and correlation.

The mean shares the total out evenly. Add up every value, then divide by how many values there are. The fair share.

The median is the middle value, once the data is in order. Order first, always. That word once is where the marks go.

The mode is the most common value. The one that appears most often. Mode, most. Same first two letters.

And the range is not an average at all. It's the spread: how far apart the biggest and the smallest are. Examiners see it offered as an average every single year. It isn't one.

So. Here's our data. Seven students, one minute each, counting press-ups. Twelve, seven, twenty, seven, fifteen, nine, seven. This one list will answer all four questions, without a single number changing.

Mean first. Add them all up. Twelve, seven, twenty, seven, fifteen, nine, seven. That comes to seventy-seven.

Seventy-seven, shared between seven students. Seven elevens are seventy-seven, so the mean is eleven press-ups each.

Now the median: the middle value. Seven numbers, so the middle one is the fourth. And the fourth number in this list is seven. Hold on to that, because here is what an examiner wrote after marking a real Foundation paper. Quote: it was fairly common to see the middle number being chosen from the unordered list. Seven is wrong.

Because the median is the middle of the ordered data. So before you point at anything, put the list in order, smallest to largest. Watch.

Seven, seven, seven, nine, twelve, fifteen, twenty. Fourth from the left, fourth from the right. The median is nine. Not the seven we nearly wrote down.

The mode is easy now the list is ordered, because equal values sit together. Seven appears three times; everything else appears once. So the mode is seven.

And the range: largest take away smallest. Twenty minus seven is thirteen. Thirteen press-ups between the strongest and the weakest. And say this one with me. The range is not an average. It is spread. Hand thirteen to an examiner who asked for an average, and you get nothing.

One list. Mean eleven, median nine, mode seven, range thirteen. Four different numbers, and every one of them is right, for the question it answers.

Your turn. A cafe sells six coffees one morning: three pounds, three pounds fifty, four pounds, three pounds, four pounds fifty, and six pounds. The worksheet just says, find the average price. Work out the mean, and the median. Pause here if you need longer.

The mean: the six prices add up to twenty-four pounds. Twenty-four divided by six is four pounds.

The median: in order, they are three, three, three fifty, four, four fifty, six. Six values, so there is no single middle one. The middle two are three fifty and four, and halfway between them is three pounds seventy-five.

Four pounds, and three pounds seventy-five. Same six coffees. The one six-pound coffee drags the mean upwards, but it cannot drag the middle position anywhere. That is why an exam never just says the average. It says mean, or median, and it means it.

Last thing, and it is not a calculation: the names. An examiner on another board wrote, quote, a small number of candidates confused median with mean. So: I'll give you the job, you name the average. The value that appears most often. The middle of the ordered list. The fair share of the total. And the one that isn't an average at all. Go.

Most often is the mode. Middle of the ordered list is the median. Fair share of the total is the mean. And the odd one out is the range: not an average, but the spread.

That was the first video of ten in the chapter. Next: finding the modal class.

So: sort before you take the middle, and always check which word the question actually used — because whoever picks the average picks the story. For more, visit scholafly.com, or watch the next video.