When we use the word "**average**", we could mean the mean, the median or the mode because these are all "averages" or more correctly, **measures of central tendency**.

**Mean** - Add up all the data scores and then divide by the number of scores there are.

**Median** - Write all the scores from smallest to largest and then choose the score in the middle.

**Mode** - This is the most frequent score (or *m*ost *o*ften score).

The population numbers of alligators in 7 national parks is below. Calculate the mean, the median and the mode.

Data - 14, 13, 8, 19, 13, 24, 15

**Answer:**

Mean | = (14 + 13 + 8 + 19 + 13 + 24 + 15) ÷ 7 |

= 106 ÷ 7 | |

= 15.1 alligators |

Ordered Data - 8, 13, 13, **14**, 15, 19, 24

Median (middle score in ordered data) = 14 alligators

Mode (*m*ost *o*ften score) = 13 alligators

The number of graffiti convictions for 12 months of monthy data is below. Calculate the mean, the median and the mode.

Data - 14, 25, 23, 15, 16, 16, 11, 23, 12, 13, 24, 7

**Answer:**

Mean | = (14 + 25 + 23 + 15 + 16 + 16 + 11 + 23 + 12 + 13 + 24 + 7) ÷ 12 |

= 199 ÷ 12 | |

= 16.6 convictions |

Ordered Data - 7, 11, 12, 13, 14, 15, 16, 16, 23, 23, 24, 25 | |

Median (middle score in ordered data) | = halfway between 15 and 16 |

= (15 + 16) ÷ 2 | |

= 15.5 convictions |

Mode = 16 and 23

There are *two* modes. The data is *bi*modal.

An **outlier** is a score which is considerably different from the rest of the data. (For example, one of your classmates may have
over 1000 Facebook "Friends"!!) Since the outlier will skew the data analysis, the outlier is often left out. However, in this example,
we will include it to show you the **skewing effect of the outlier on the mean**.

The number of Facebook "Friends" is below. Calculate the mean, the median and the mode.

Data - 12, 15, 3, 10, 25, 1000, 12

**Answer:**

Mean | = (12 + 15 + 3 + 10 + 25 + 1000 + 12) ÷ 7 |

= 1077 ÷ 7 | |

= 153.9 Friends!! |

Ordered Data - 3, 10, 12, **12**, 15, 25, 1000

Median = 12 Friends

Mode = 12 Friends

The mean, median and mode are all equal for the set of numbers [ 3, 4, 5, 8, X ]. What is the value of X?

**Q1.** Numbers of hot dogs eaten by contestants in a ten-minute competition - 63, 15, 10, 12, 23

**Q2.** Liver organ transplant survival times (years) - 16, 15, 23, 25, 28, 12, 12, 18

**Answers**

A1. Mean = 24.6 | Median = 15 | Mode = no mode |

A2. Mean = 18.6 | Median = 17 | Mode =12 |

In statistics, the data is usually so large that it must be organized into tables with extra columns. The rules to calculate the mean and the median are adapted to suit this tabular format.

Mean of Organized Data | ||

Mean = | ∑ fx | (where Sigma ∑ means "the sum of") |

∑ f |

**Median of Organized Data**

The median score is halfway through the data scores so the median is identified using the rule:

( ∑ f + 1 ) |

2 |

where

∑ *f* is the the sum of the frequencies. Refer to the examples below.

Calculate the mean, the median and the mode from this road accident fatalities data:

AGE (YEARS) | NUMBER OF DEATHS |

16 | 5 |

17 | 8 |

18 | 20 |

19 | 27 |

**Answer:**

AGE (YEARS)x | NUMBER OF DEATHSf | f x |

16 | 5 | 16 × 5 = 80 |

17 | 8 | 17 × 8 = 136 |

18 | 25 | 18 × 25 = 450 |

19 | 27 | 19 × 27 = 513 |

∑ f = 65 | ∑ fx = 1179 |

Mean = | ∑ fx | = | 1179 | = 18.14 years |

∑ f | 65 |

Median score = | (∑ f + 1) | = | (65 + 1) | = 33rd score = 18 years |

2 | 2 |

*(The data has already been ordered from smallest to largest in the table.
Count down the frequency column to find the 33rd score.
The 33rd person is in the row for the 18 years' age group.)*

Mode = 19 years

According to the Guinness World Records, an American woman, Lee Redmond, took 30 years to grow all ten fingernails with an "average" length of 75 centimetres. Sadly, she lost all her fingernails in a car accident in 2008. Happily, she suffered no other serious injuries.

Real estate agents and property developers only use the **median** house price.

However, in this example of house prices, we will work out the mean, the median and the mode.

HOUSE PRICES ($) | NUMBER OF HOUSES |

100 000 - 199 999 | 10 |

200 000 - 299 999 | 12 |

300 000 - 399 999 | 13 |

**Answer:**

HOUSE PRICES ($) CLASS | HOUSE PRICES ($) MIDPOINTx |
NUMBER OF HOUSESf | f x |

100 000 - 199 999 | 149 999.50 | 10 | 1 499 995.00 |

200 000 - 299 999 | 249 999.50 | 12 | 2 999 994.50 |

300 000 - 399 999 | 349 999.50 | 13 | 4 549 993.50 |

∑ f = 35 | ∑ fx = 9 049 983.00 |

Mean = | ∑ fx | = | 9 049 983.00 | = 258 570.94 = $ 258 600 approx. |

∑ f | 35 |

Median score = | (∑ f + 1) | = | (35 + 1) | = 18th house price = $200 000 to $299 999 |

2 | 2 |

*(The data has already been ordered from smallest to largest in the table.
Count down the frequency column to find the 18th score.
The 18th house price is in the 2nd row.)*

Mode = $300 000 to $399 999

According to the Guinness World Records, the **longest hair** grown by a Chinese woman, Xie Qiuping, measured 5.627 metres long.
If the "average" hair growth is about 15 centimetres per year, how long did this take?