# The Mean from a Frequency Table

It is easy to calculate the Mean:

**Add up** all the numbers,

then **divide by how many** numbers there are.

### Example: What is the Mean of these numbers?

6, 11, 7

- Add the numbers:
**6 + 11 + 7 = 24** - Divide by
*how many*numbers (there are 3 numbers):**24 ÷ 3 = 8**

**The Mean is 8**

But sometimes we don't have a simple list of numbers, it might be a frequency table like this (the "frequency" says how often they occur):

Score | Frequency |
---|---|

1 | 2 |

2 | 5 |

3 | 4 |

4 | 2 |

5 | 1 |

*(it says that score 1 occurred 2 times, score 2 occurred 5 times, etc)*

We could list all the numbers like this:

Mean = \frac{1+1 + 2+2+2+2+2 + 3+3+3+3 + 4+4 + 5}{(how many numbers)}

But rather than do lots of adds (like 3+3+3+3) it is easier to use multiplication:

Mean = \frac{2×1 + 5×2 + 4×3 + 2×4 + 1×5}{(how many numbers)}

And rather than count how many numbers there are, we can add up the frequencies:

Mean = \frac{2×1 + 5×2 + 4×3 + 2×4 + 1×5}{2 + 5 + 4 + 2 + 1}

And now we calculate:

Mean = \frac{2 + 10 + 12 + 8 + 5}{14}

= \frac{37}{14} = **2.64...**

And that is how to calculate the mean from a frequency table!

Here is another example:

### Example: Parking Spaces per House in Hampton Street

Isabella went up and down the street to find out how many parking spaces each house has. Here are her results:

Parking Spaces | Frequency |
---|---|

1 | 15 |

2 | 27 |

3 | 8 |

4 | 5 |

What is the mean number of Parking Spaces?

Answer:

The Mean is **2.05** (to 2 decimal places)

*(much easier than adding all numbers separately!)*

## Notation

Now you know how to do it, let's do that last example again, but using formulas.

This symbol (called Sigma) means "sum up" (read more at Sigma Notation) |

So we can say "add up all frequencies" this way:

(where

*f*is frequency)

And we can use it like this:

Likewise we can add up "frequency times score" this way:

(where *f* is frequency and *x* is the matching score)

And the formula for calculating the mean from a frequency table is:

*The x with the bar on top says "the mean of x"*

So now we are ready to do our example above, but with correct notation.

### Example: Calculate the Mean of this Frequency Table

x | f |
---|---|

1 | 15 |

2 | 27 |

3 | 8 |

4 | 5 |

And here it is:

x = \frac{Σfx}{Σf} = \frac{15×1 + 27×2 + 8×3 + 5×4}{15+27+8+5}

= **2.05...**

There you go! You can use sigma notation.

## Calculate in the Table

It is often better to do the calculations **in** the table.

### Example: (continued)

From the previous example, calculate *f × x* in the right-hand column and then do totals:

x | f | fx |
---|---|---|

1 | 15 | 15 |

2 | 27 | 54 |

3 | 8 | 24 |

4 | 5 | 20 |

TOTALS: | 55 | 113 |

And the Mean is then easy:

Mean = \frac{113}{55} = **2.05...**