We know that we can use graphs to show statistical information in a meaningful way that people can look at and quickly understand. Pie charts are useful for showing things like how many votes each candidate in an election is likely to get. Line charts are great for showing information over time, such as the temperature over an entire year.

There are other situations where graphs aren’t necessarily the best way of presenting the information. Say you looked at the ages of all the kids in your entire grade. There might be 100 kids in your grade, and if you listed their ages you might get this:

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15, 14, 14, 15, 15, 15, 15, 16, 15, 14, 15, 15, 15… and so on

There isn’t really any easy way to *graph*
this information. We have to resort to other ways of talking about the
information.

There are three main *measures* used to give
people a ‘rough idea’ about all the data. These three measures are called the *mean,
median and mode*. To show how each of these measures works, we’ll use the
ages of an example class of 13 students:

15, 14, 14, 15, 15, 15, 15, 16, 15, 14, 15, 15, 15

### Mean

The *mean* is the *average* of all the
data. There is a special symbol for the mean of a set of numbers – it’s an ‘x’
with a horizontal line above it: _{}. The ‘x’ part represents the bunch
of numbers, and the horizontal line means ‘the average of’.

To find the mean of the ages in our class, you need to add up all the ages, and then divide this sum by the number of kids in the class. In mathematical terms, this is:

_{}

There’s a special symbol that tells you to add up all the numbers, it’s called a ‘sigma’ symbol. Instead of writing ‘sum of all the numbers’, you can use the sigma symbol, like this:

_{}

Remember that the ‘x’ just represents all the numbers, and the ‘sigma’ symbol to the left of it just tells you to add up all the numbers represented by ‘x’.

Also, instead of writing, “How many numbers there are,” often in statistics you can represent how many numbers there are by the letter ‘n’. Think of ‘n’ standing for the (n)umber of numbers. So we can rewrite our formula for finding the average like this:

_{}

So, let’s use this formula to find the mean or average age of the kids in the class:

_{}

Is this what we expect? Well, the youngest kids in the class are 14 years old, and the oldest kid is 16 years old. Most of the kids are 15 years old. There are more 14 year old kids than there are 16 year old kids, so the average age is probably going to be somewhere around 15 years, but probably a bit less. 14.85 agrees with what we expect.

### Median

To get the *median* of a set of numbers, you
first need to write the numbers in order from smallest to largest, or vice
versa. So, let’s order the ages in our class from youngest to oldest:

14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16

The median value is the *middle* value in the
list once you’ve ordered them. To quickly find the median value, follow this
procedure:

· Count up how many values you have in your list

·
If there are an *even* number of values, divide the number
by two. If there are an *odd* number of values, add one to the number *and
then* divide it by 2.

·
Take the answer from part 2 and count that many along the list
starting from the left. So the leftmost number counts as 1, the next number
counts as 2, and so on. If there is an even number of values, the *average *of
the number you arrive at *and the number to its right* is the median
value. If there an *odd* number of values, the number you arrive at is
the median value.

Let’s use this procedure to find the median of the ages in our class:

There are **13**
values in our list.

Since we have an *odd* number of values, we need
to add 1 to it and then divide by 2:

13 + 1 = 14

14
÷ 2 = __7__

We need to count 7 numbers across from the left hand number, which count as 1:

Because we have an odd number of values, the median value is just the number we arrived at after counting 7 across.

What if we’d had an *even* number of ages –
say we’d only had 12 in our list (I got rid of one of the ‘15’s):

14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16

Then the procedure would have been like this:

There are **12**
values in our list.

Since we have an *even* number of
values, we just need to divide it by 2:

12
÷ 2 = __6__

We need to count 6 numbers across from the left hand number:

Because we have an *even* number of values, we
have to look at the average of two values. They are the number we arrived at
after counting 6 across, and the number just to its right – 15 and 15. The
average of 15 and 15 is 15! So the median is 15.

### Mode

The mode is the most commonly occurring number in the entire list. Usually, there is only one mode in a group of numbers, but sometimes you can get more than one. Often, the mode is really obvious, but to find it for sure, you need to draw up a table and count how many times each number occurs. For our class of 13 students, the table would look like this:

For ages 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16:

Student age |
Number of students with that age |

14 |
3 |

15 |
9 |

16 |
1 |

‘15’ is the most commonly occurring age, so 15 is
the *mode* for this set of numbers. What about if for a different class
of students, our table looked like this:

Student age |
Number of students with that age |

14 |
6 |

15 |
6 |

16 |
1 |

In this case, ages ‘14’ and ‘15’ both occur the
same number of times – 6 times each. This means that we have *two modes*
– 14 *and *15.