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# Measures of Central Tendency and Dispersion

## Mean, median, mode, range

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Measures of Central Tendency and Dispersion

### 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 mean=11+16+9+15+5+186=746=1213\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

19630+28920+56330+49930+69030+39130+35460+24765908=346,877.50\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. License: CC BY-NC 3.0 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 1+3+4+5+6+7+3+1=30\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: 2(1)+3(3)+4(4)+5(5)+6(6)+7(7)+8(3)+9(1)30=17030=523\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 15th\begin{align*}15^{th}\end{align*} and 16th\begin{align*}16^{th}\end{align*} oldest (that way there will be 15 younger and 15 older than the median age). Both the 15th\begin{align*}15^{th}\end{align*} and 16th\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. Mean=2+17+1+(3)+12+8+12+169=7.22¯¯¯¯¯\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 1. Find the median and mode of the numbers given in Example A, specifically: 11, 16, 9, 15, 5, and 18. 2. 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. 3. Find the mean, median and mode of the data set: 14, 9, 3, 14, 2, 7, 13, 6. 4. Find the mean, median and mode of the data set: 5, 3, 5, 0, 1, 5, 3, 4, 4, 4. 5. 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. 6. 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 7. 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.

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

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

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### Vocabulary Language: English

TermDefinition
arithmetic mean The arithmetic mean is also called the average.
descriptive statistics In descriptive statistics, the goal is to describe the data that found in a sample or given in a problem.
inferential statistics With inferential statistics, your goal is use the data in a sample to draw conclusions about a larger population.
measure of central tendency In statistics, a measure of central tendency of a data set is a central or typical value of the data set.
Median The median of a data set is the middle value of an organized data set.
Mode The mode of a data set is the value or values with greatest frequency in the data set.
multimodal When a set of data has more than 2 values that occur with the same greatest frequency, the set is called multimodal    .
Outlier In statistics, an outlier is a data value that is far from other data values.
Population Mean The population mean is the mean of all of the members of an entire population.
resistant A statistic that is not affected by outliers is called resistant.
Sample Mean A sample mean is the mean only of the members of a sample or subset of a population.