### Measures of Central Tendency

The word “average” is often used to describe the general characteristics of a group of unequal objects. Mathematically, an average is a single number which can be used to summarize a collection of numerical values. In mathematics, there are several types of “averages” with the most common being the **mean**, the **median** and the **mode**.

**Mean**

The **arithmetic mean** of a group of numbers is found by dividing the sum of the numbers by the number of values in the group. In other words, we add all the numbers together and divide by the number of numbers.

#### Finding the Mean

Find the mean of the numbers 11, 16, 9, 15, 5, 18.

There are six separate numbers, so the \begin{align*}\text{mean} = \frac{11 + 16 + 9+ 15+ 5+18}{6}=\frac{74}{6}=12\frac{1}{3}\end{align*}.

The arithmetic mean is what most people automatically think of when the word average is used with numbers. It’s generally a good way to take an average, but it can be misleading when a small number of the values lie very far away from the rest. A classic example would be when calculating average income. If one person (such as former Microsoft Corporation chairman Bill Gates) earns a great deal more than everyone else who is surveyed, then that one value can sway the mean significantly away from what the majority of people earn.

#### Real-World Application: Annual Income

The annual incomes for 8 professions are shown below. Form the data, calculate the mean annual income of the 8 professions.

Profession |
Annual Income |
---|---|

Farming, Fishing, and Forestry | $19,630 |

Sales and Related | $28,920 |

Architecture and Engineering | $56,330 |

Healthcare Practitioners | $49,930 |

Legal | $69,030 |

Teaching & Education | $39,130 |

Construction | $35,460 |

Professional Baseball Player* | $2,476,590 |

(*Source:* Bureau of Labor Statistics, except (*)-The Baseball Players' Association (playbpa.com)).

There are 8 values listed, so the mean is

\begin{align*}\frac{19630+28920+56330+49930+69030+39130+35460+2476590}{8}= \$346,877.50\end{align*}

As you can see, the *mean* annual income is substantially larger than the income of 7 out of the 8 professions. The effect of the single **outlier** (the baseball player) has a dramatic effect on the mean, so the mean is not a good method for representing the ‘average’ salary in this case.

**Median**

The median is another type of average. It is defined as the value in the middle of a group of numbers. To find the median, we must first list all the numbers *in order from least to greatest*.

#### Finding the Median

Find the median of the numbers 11, 21, 6, 17, 9.

We first list the numbers in ascending order: 6, 9, **11**, 17, 21.

The median is the value *in the middle* of the set (in bold).

**The median is 11**. There are two values higher than 11 and two values lower than 11.

If there is an even number of values, then the median is the arithmetic mean of the two numbers in the middle (in other words, the number halfway between them).

The median is a useful measure of average when the data set is highly skewed by a small number of points that are extremely large or extremely small. Such outliers will have a large effect on the mean, but will leave the median relatively unchanged.

**Mode**

The mode can be a useful measure of data when that data falls into a small number of categories. It is simply a measure of the most common number, or sometimes the most popular choice. The mode is an especially useful concept for data sets that contains non-numerical information, such as surveys of eye color or favorite ice-cream flavor.

Of course, a data set can contain more than one mode; when it does, it is called **multimodal**. In fact, every value in a data set could be a mode, if every value appears an equal number of times. However, this situation is quite rare. You might encounter data sets with two or even three modes, but more than that would be unlikely unless you are working with very small sample sets.

#### Real-World Application: Age of Customers

Jim is helping to raise money at his church bake sale by doing face painting for children. He collects the ages of his customers, and displays the data in the graph below. Find the mean, median and mode for the ages represented.

By reading the graph we can see that there was one 2-year-old, three 3-year-olds, four 4-year-olds, etc. In total, there were \begin{align*}1 + 3 + 4 + 5 + 6 + 7 + 3 + 1 = 30\end{align*} customers.

The mean age is found by adding up all the ages multiplied by the number of times each age appears, and then dividing by 30:

\begin{align*}\frac{2(1)+3(3)+4(4)+5(5)+6(6)+7(7)+8(3)+9(1)}{30} = \frac{170}{30} = 5 \frac{2}{3} \end{align*}

Since there are 30 children, the median is half way between the \begin{align*}15^{th}\end{align*} and \begin{align*}16^{th}\end{align*} oldest (that way there will be 15 younger and 15 older than the median age). Both the \begin{align*}15^{th}\end{align*} and \begin{align*}16^{th}\end{align*} oldest fall in the 6-year-old range, therefore the median is 6.

The mode is given by the age group with the highest frequency. Reading directly from the graph, we see that the mode is 7; there are more 7-year-olds than any other age.

Watch this video for assistance with the example above:

### Example

#### Example 1

Find the mean, median and mode of the numbers 2, 17, 1, -3, 12, 8, 12, 16.

\begin{align*}\text{Mean}=\frac{2+17+1+(-3)+12 + 8 + 12 +16}{9}= 7.\overline{22}\end{align*}

We first list the numbers in ascending order: -3, 1, 2, 8, 12, 12, 16, 17.

The median is the value *in the middle* of the set, so the median lies between 8 and 12. Halfway between 8 and 12 is 10, so **10 is the median**.

The **mode** is the most frequent number or numbers. The only number that repeats is 12, so **12 is the mode.**

### Review

- Find the
**median**and**mode**of the numbers given in Example A, specifically: 11, 16, 9, 15, 5, and 18. - Find the
**median**and**mode**of the salaries given in Example B, specifically: $19,630, $28,920, $56,330, $49,930, $69,030, $39,130, $35,460, and $2,476,590. - Find the mean, median and mode of the data set: 14, 9, 3, 14, 2, 7, 13, 6.
- Find the mean, median and mode of the data set: 5, 3, 5, 0, 1, 5, 3, 4, 4, 4.
- Find the mean, median and mode of the data set: 8, 5, 10, 4, 4, 10, 6, 4, 7, 8, 2, 8, 10, 9, 2, 1, 6, 10, 5, 3.
- Find the mean, median and mode of the following numbers. Which of these will give the best
*average*? 15, 19, 15, 16, 11, 11, 18, 21, 165, 9, 11, 20, 16, 8, 17, 10, 12, 11, 16, 14 - Ten house sales in Encinitas, California are shown in the table below. Find the mean, median and mode of the sale prices. Explain, using the data, why the
**median house price**is most often used as a measure of the house prices in an area.

Address |
Sale Price |
Date Of Sale |
---|---|---|

643 3RD ST | $1,137,000 | 6/5/2007 |

911 CORNISH DR | $879,000 | 6/5/2007 |

911 ARDEN DR | $950,000 | 6/13/2007 |

715 S VULCAN AVE | $875,000 | 4/30/2007 |

510 4TH ST | $1,499,000 | 4/26/2007 |

415 ARDEN DR | $875,000 | 5/11/2007 |

226 5TH ST | $4,000,000 | 5/3/2007 |

710 3RD ST | $975,000 | 3/13/2007 |

68 LA VETA AVE | $796,793 | 2/8/2007 |

207 WEST D ST | $2,100,000 | 3/15/2007 |

For 8-10, determine which measure of central tendency (mean, median, or mode) would be most appropriate for the following.

- The life expectancy of store-bought goldfish.
- The age in years of audience for a kids TV program.
- The average actual weight of sacks of potatoes labeled as 5-pound bags.

### Review (Answers)

To view the Reviwe answers, open this PDF file and look for section 13.9.