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# Mode

## Number or numbers occurring the most often.

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Mode - Probability and Statistics

Noela is working on a homework assignment for her social studies class, and needs to find the 10-year historical period between 1900 and 2000 with the greatest number of recorded hurricanes worldwide. If she uses a data sheet listing all recorded hurricanes, what measure of central tendency would she use to identify the decade with the most hurricanes?

Credit: Chuck Simmins
Source: https://www.flickr.com/photos/chucksimmins/2859050569

### The Mode

The mode is the value(s) in a set that occurs with the greatest frequency. Of the three common measures of central tendency, the mode is the only one that may actually be one of the extremes in a set with more than one value. In certain circumstances, the mean of a set with differing values may approach one of the extremes, but only the mode may actually be one of them.

Identification of the mode(s) is simple:

• Organize the set in numerical order (to make it easier to count repeating values) and make note of the frequencies of any repeated values (any values with a frequency greater than 1)
• The value(s) occurring with the greatest frequency are the mode(s)

Because the mode is not directly related to the middle position in the organized series of values, if there are multiple values with the same frequency, do not be concerned if there is a large difference between different modes.

A set with only one mode is called a unimodal set. A set with two modes is a bimodal set. Technically, there are also trimodal sets, but generally any more than two modes are simply referred to as multimodal.

#### Identifying the Mode

Identify the mode of \begin{align*}z\end{align*}.

\begin{align*}z=\left \{3, 5, 13, 18, 3, 7, 9, 12, 11, 3, 9, 5, 4, 3, 13\right \}\end{align*}If we put the values in ascending order, we get:

3, 3, 3, 3, 4, 5, 5, 7, 9, 9, 11, 12, 13, 13, 18

Since 3 is the only value that appears four times, and all other values appear 3 or fewer times, 3 is the mode of \begin{align*}z\end{align*}.

#### Identifying the Mode Given a Histogram

Identify the mode of the set described by the histogram (the mode of the number of reported pranks):

The intervals 1985-1990 and 1995-2000 are the only two with matching frequencies: 172 reported pranks.

The mode is 172 reported pranks

#### Looking at Frequency Polygons

Answer the questions using the frequency polygons jointly graphed in the image below.

a. Which years demonstrate unimodal distribution?

Which years demonstrate unimodal distribution? 2007 has only one mode: 30-40

b. Which years demonstrate bimodal distribution?

Which years demonstrate bimodal distribution? 2008 has two modes: 30-40 and 50-60

c. Which years demonstrate multimodal distribution OR have no mode?

Which years demonstrate multimodal distribution OR have no mode? 2006 has 3 modes: 20-30, 40-50, 70-80

d. Which year(s) have the mode with the greatest frequency?

Which year(s) have the mode with the greatest frequency? Year 2008 has a mode of 25, the greatest on the chart.

#### Earlier Problem Revisited

Noela is working on a homework assignment for her social studies class, and needs to find the 10-year historical period between 1900 and 2000 with the greatest number of recorded hurricanes worldwide. If she uses a data sheet listing all recorded hurricanes, what measure of central tendency would she use to identify the decade with the most hurricanes?

Noela needs to organize the data by decade, then identify the mode, this will be the decade with the greatest frequency of hurricanes.

### Examples

Find the mode:

#### Example 1

\begin{align*}\left \{12, 17, 12, 63, 17, 12, 54, 23, 39\right \}\end{align*}

Put the values in numerical order (not truly necessary, but makes it easier to identify multiples): 12, 12, 12, 17, 17, 23, 39, 54, and 63.

There are three 12’s (highlighted in red), two 17’s, and only one of each other value. 12 is the mode.

#### Example 2

\begin{align*}\begin{array}{c|c c c c c c c c c c c c c c c c c c c} 6 & 7 & 7 & 9 \\ 7&3&4&4&6&8&9\\ 8&0&1&2&3&3&3&3&4&4&6&7&7\\ 9&0\end{array}\end{align*}

This stem plot already lists the values in ascending order, so it is easy to see that 83 is the mode, with a frequency of 4.

#### Example 3

\begin{align*}\left \{93, 91, 95, 92, 92, 93, 95, 94, 97, 93, 86, 92, 94, 89, 92, 91, 92, 93, 94, 100\right \}\end{align*}

If we put the values in ascending order: 86, 89, 91, 91, 92, 92, 92, 92, 92, 93, 93, 93, 93, 94, 94, 94, 95, 95, 97, and 100, we can see right away that 92 is the mode, with frequency 5.

#### Example 4

\begin{align*}\left \{275, 281, 269, 280, 268, 278, 279, 274, 275, 281, 285, 285, 278, 269, 283, 263, 277, 276, 269, 281, 272, 275, 276\right \}\end{align*}

In numerical order, the set looks like: 263, 268, 269, 269, 269, 272, 274, 275, 275, 275, 276, 276, 277, 278, 278, 279, 280, 281, 281, 281, 283, 285, and 285. It is apparent now that this is a multimodal set with modes: 269, 275, and 281.

### Review

Find the mode(s):

1. \begin{align*}\left \{326, 314, 325, 315, 315, 307, 318, 318, 320, 312, 325, 321, 312, 320, 312, 325, 326, 325\right \}\end{align*}
2. \begin{align*}\left \{35, 37, 28, 42, 32, 42, 35, 45, 28, 43, 37, 43, 27, 41, 27, 45, 31, 42, 28, 45\right \}\end{align*}
3. \begin{align*}\left \{123, 167, 150, 167, 152, 128, 129, 150, 140, 121\right \}\end{align*}
4. \begin{align*}\left \{2120, 3040, 2180, 1892, 923, 9231, 8231\right \}\end{align*}
5. \begin{align*}\left \{12, 23, 41, 23, 61, 130, 210, 130, 592, 130, 12\right \}\end{align*}
6. \begin{align*}\left \{{23.43, 32.52, 23.92, 32.25, 23.43, 29.55, 28.30, 23.34}\right \}\end{align*}
7. \begin{align*}\left \{\frac{1}{2}, \frac{4}{9}, \frac{3}{7}, \frac{2}{5}, \frac{21}{23}, \frac{16}{27}, \frac{2}{4}\right \}\end{align*}
8. \begin{align*}\left \{{.57, \frac{23}{100}, .44, \frac{17}{100}, \frac{52}{100}, .23, .44, \frac{45}{100}}\right \}\end{align*}
9. \begin{align*}\left \{12, 1.2, .012, 102, 120, .012, .12, 1.2\right \}\end{align*}
10. \begin{align*}\left \{123, 12.3, \frac{12}{30}, \frac{123}{100}, \frac{120}{123}, \frac{30}{12}, \frac{1}{123}, 1.23, .123\right \}\end{align*}
11. \begin{align*}\begin{array}{c|c c c c c c c c c c c c c c c c c c c} 2 & 2 & 2\\ 3&3&5&5&6&9&9&9\\ 4&1&2&2&3&3&4&4&8\\ 5&2&4&5&5&7&7&7\end{array}\end{align*}

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### Vocabulary Language: English

TermDefinition
bimodal If there are two numbers that occur equally frequently in a set of data, then the data is said to be bimodal.
Frequency In statistics, the frequency of an event is the number of times that event occurred.
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    .
normal distribution curve A normal distribution curve is a symmetrical curve that shows the highest frequency in the center with an identical curve on either side of the center.
trimodal A trimodal set has three modes and may be also referred to as multimodal.
unimodal If a data set has only 1 value that occurs most often, the set is called  unimodal.