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Measures of center are used to give an idea of a *typical* value of a data set. Instead of presenting all the data points, a measure of center can be used to represent the entire set.

mean $=number of valuessum of values $

Suppose a data set represents the heights of different towers. The mean of this data set gives an idea of a **typical** height. Calculating the mean could be seen as rearragning the blocks so that all the towers have the same height.

Animate the mean value

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After the blocks are rearranged, the towers each have a height of 4. Therefore, the mean is 4. If the heights are written as x, then the mean is sometimes written as $xˉ.$ The towers' mean height can then be written as $xˉ=4.$

4,3,1,2,4,4,5,4,

the mode is 4, since it occurs the most number of times. When there are two or more values that are just as common, there can be more than one mode. However, if all values only occur once, then the data set does not have a mode.
Find the mean, median and the mode for the following data.

Show Solution

We'll find each statistic one at a time.
### Example

### Mean

In order to determine the mean, we need to add all the data points together. Then we divide the sum by the total number of points, which, in this case, is 10.
The mean is 5.9.
### Example

### Median

To determine the median, we first write the data points in ascending order.
The median is the value that lies directly in the middle. Since there is an even number of values, there is no single middle number. The median is then found by calculating the mean of the two middle numbers, 5 and 7.
### Example

### Mode

The mode is the value that is most common in the data set. In this case, it is 9, which occurs three times. Therefore, the mode is 9.

$median=25+7 =6.$

The median is 6.
There isn't anything Benji loves more than a bumble bee. Every day he keeps track of how many he sees. The data below gives the number of bumble bees Benji saw for the first 13 days in April.
On the 14th day, Benji sees 7 bumble bees! Determine how this value affects the mean, median and mode of the data.

Show Solution

To begin, we should notice that 7 bumble bees is significantly higher than the number of bees Benji saw on any other day. Thus, 7 is probably an outlier. To see how this outlier affects the measures of center, we can calculate all three with and without the outlier.

$meanmedianmode :1.85:22:21 →2.38→22→21 $

Adding an additional data point, even when it was such an extreme data point, only affected the mean. The median and mode remained the same. This is because the mean is the only measure of center that takes into account the actual data points rather than just the number of data points. Thus, the outlier affects the mean.
Each of the three measures of center, mean, median, and mode, has advantages and drawbacks.

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