4.4 Measures of central tendency
4.4.three Calculating the way

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When it's unique, the mode is the value that appears the most often in a data set and information technology tin can exist used as a measure of fundamental tendency, like the median and mean. But sometimes, there is no fashion or there is more than than one style.

There is no fashion when all observed values appear the aforementioned number of times in a data prepare. There is more than one fashion when the highest frequency was observed for more than one value in a data set. In both of these cases, the fashion tin can't exist used to locate the centre of the distribution.

The mode tin be used to summarize categorical variables, while the mean and median tin be calculated only for numeric variables. This is the main advantage of the way as a measure of key tendency. It's besides useful for discrete variables and for continuous variables when they are expressed equally intervals.

Here are some examples of calculation of the mode for discrete variables.

Example one – Number of points during a hockey tournament

During a hockey tournament, Audrey scored 7, 5, 0, 7, viii, five, 5, 4, ane and 5 points in x games. After summarizing the information in a frequency table, y'all tin can easily come across that the manner is 5 because this value appears the most frequently in the information set (4 times). The mode can be considered a measure of fundamental tendency for this data set because it's unique.



Table 4.4.three.i
Number of games by the number of points scored
Tabular array summary
This table displays the results of Number of games past the number of points scored. The information is grouped past Number of points scored (appearing every bit row headers), Frequency (number of games) (actualization as column headers).
Number of points scored Frequency (number of games)
0   i
1   i
4   i
5   4
7   two
8   1
0 truthful zero or a value rounded to zero

Example 2 – Number of points in 12 basketball games

During Marco's 12-game basketball game flavor, he scored xiv, 14, xv, 16, 14, sixteen, 16, 18, xiv, 16, xvi and fourteen points. Later on summarizing the information in a frequency table, you tin run across that there are two modes in this data fix: 14 and 16. Both values appear 5 times in the data set up and five is the highest frequency observed. The mode tin't be used a measure out of cardinal trend because there is more than ane fashion. It'southward a bimodal distribution.



Table 4.4.3.2
Number of games by the number of points scored
Table summary
This table displays the results of Number of games by the number of points scored. The information is grouped past Number of points scored (appearing as row headers), Frequency (number of games) (appearing as column headers).
Number of points scored Frequency (number of games)
14 5
15 ane
xvi v
18 one

Example 3 – Number of touchdowns scored during football season

The post-obit data set up represents the number of touchdowns scored past Jerome in his loftier-school football game season: 0, 0, 1, 0, 0, 2, 3, i, 0, 1, 2, 3, 1, 0. Allow'due south compare the mean, median and mode.
The sum of all values is 14 and there are xiv data points. This gives a mean of one. Because the number of values is even, the median is average betwixt the data indicate of rank seven and the information point of rank viii, after arranging the data set in increasing order.



Table 4.iv.3.three
Rank associated with each value of the number of touchdowns during football season
Table summary
This table displays the results of Rank associated with each value of the number of touchdowns during football season. The information is grouped by Rank (actualization as row headers), Number of touchdowns (appearing as column headers).
Rank Number of touchdowns
1 0
2 0
3 0
4 0
five 0
6 1
vii one
eight 1
ix 1
10 1
11 2
12 2
13 iii
14 3

Therefore, the median is equal to 1. Once the information has been summarized in a frequency table, you tin can see that the mode is 0 because it is the value that appears the virtually often (vi times).



Table 4.4.3.4
Number of games by the number of touchdowns
Tabular array summary
This table displays the results of Number of games by the number of touchdowns. The information is grouped by Number of touchdowns (appearing as row headers), Frequency (appearing every bit column headers).
Number of touchdowns Frequency
0 6
1 4
2 2
3 2
0 true naught or a value rounded to zippo

In summary, in this example, the mean is 1, the median is one and the mode is 0.

The mode is not used every bit much for continuous variables because with this type of variable, it is likely that no value will appear more than once. For example, if you ask xx people their personal income in the previous twelvemonth, information technology's possible that many will have amounts of income that are very close, but that yous will never get exactly the same value for two people. In such case, it is useful to grouping the values in mutually exclusive intervals and to visualize the results with a histogram to identify the modal-class interval.

Example 4 – Height of people in the arena during a basketball game

We are interested in the height of the people present in the arena during a basketball game game. Tabular array 4.4.3.5 presents the number of people for 20-centimetre intervals of height.



Tabular array iv.four.three.5
Number of people by acme intervals
Table summary
This table displays the results of Number of people by height intervals. The information is grouped by Height (in centimetres) (appearing as row headers), Frequency (number of people) (actualization every bit column headers).
Height (in centimetres) Frequency (number of people)
20 to 39 42
40 to 59 105
sixty to 79 176
fourscore to 99 230
100 to 119 214
120 to 139 168
140 to 159 363
160 to 179 480
180 to 200 170
200 to 219 11

Chart 4.iv.3.1 shows this data gear up as a histogram.

Chart 4.4.3.1 Histogram of the height of people at the basketball match

Data table for Chart iv.4.3.1

Information illustrated in this chart are the data from table 4.iv.iii.5.

Looking at the tabular array and histogram, you can hands identify the modal-class interval, 160 to 179 centimetres, whose frequency is 480. You lot tin can also see that equally the acme decreases from this interval, the frequency also decreases for the interval 140 to 159 centimetres (363) and it continues to subtract for 120 to 139 centimetres (168), before starting to increase until the acme reaches lxxx to 99 centimetres (230).

For categorical or discrete variables, multiple modes are values that reach the same frequency: the highest i observed. For continuous variables, all peaks of the distribution can be considered modes fifty-fifty if they don't accept the same frequency. The distribution for this example is bimodal, with a major mode corresponding to the modal-class interval 160 to 179 centimetres and a minor mode corresponding to the modal-course interval 80 to 99 centimetres. The modal form shouldn't be used as a measure of central tendency, but finding two modes gives us an indication that there could be two singled-out groups in the data that should exist analyzed separately.


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