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7.06 Outliers and measures of center

Lesson

Identifying outliers

An outlier is a data point that varies significantly from the rest of the data. An outlier will be a value that is either significantly larger or smaller than other observations. 

Exploration

Consider the dot plot below. We would call $9$9 an outlier as it is well above the rest of the data.

Practice questions

Question 1

Identify the outlier(s) in the data set $\left\{73,77,81,86,131\right\}${73,77,81,86,131}.

Question 2

Rochelle recorded the heights (in centimeters) of all the students in her class. She recorded the following:

$148,153,137,142,140,146,136,143,135,144,189,138,149,139,145,150$148,153,137,142,140,146,136,143,135,144,189,138,149,139,145,150

Identify the outlier.

 

Effect of outliers

Once we identify an outlier we should further investigate the underlying cause of the outlier.

If the outlier is a mistake then it should be removed from the data. If the data is not a mistake it should not be removed from the data set as while it is unusual it is representative of possible outcomes.

For example, you would not remove a very tall student's height from data for a class just because it was unusual for the class.

When data contains an outlier we should be aware of its impact on any calculations we make. Let's look at the effect that outliers have on the three measures of center - mean, median and mode:

Measure of center Effect of outlier
Mean

The mean will be significantly affected by the inclusion of an outlier:

  • Including a high outlier will increase the mean
  • Including a low outlier will decrease the mean
Median

The median is the middle value of a data set, the inclusion of an outlier will not generally have a significant impact on the median unless there is a large gap in the center of the data. Thus, generally:

  • Including a high outlier may increase the median slightly or it may remain unchanged
  • Including a low outlier may decrease the median slightly or it may remain unchanged
Mode

The mode is the most frequent value, as an outlier is an unusual value it will not be the mode. Hence:

  • The inclusion of an outlier will have no impact on the mode

 

Practice questions

Question 3

A set of data has a mean of $x$x, the outlier is removed and the mean rises. The outlier must have had:

  1. a value, but we cannot tell if it was larger or smaller

    A

    a value smaller than the values that remain

    B

    a value larger than the values that remain

    C

    a value, but we cannot tell if it was larger or smaller

    A

    a value smaller than the values that remain

    B

    a value larger than the values that remain

    C

Question 4

Consider the following set of data:

$37,46,35,56,56,35,125,36,48,56$37,46,35,56,56,35,125,36,48,56

  1. Fill in this table of summary statistics.

    Mean $\editable{}$
    Median $\editable{}$
    Mode $\editable{}$
  2. Which data value is an outlier?

  3. Fill in this table of summary statistics after removing the outlier.

    Mean $\editable{}$
    Median $\editable{}$
    Mode $\editable{}$
  4. Let $A$A be the original data set and $B$B be the data set without the outlier.

    Fill in this table using the symbols $>$>, $<$< and $=$= to compare the statistics before and after removing the outlier.

      With outlier Without outlier
    Mean: $A\editable{}B$AB
    Median: $A\editable{}B$AB
    Mode: $A\editable{}B$AB

 

Best choice of center

Recall that we can use the mean, median or mode to describe the center of a data set.

Sometimes one measure may better represent the data than another. When deciding which to use we need to ask ourselves "Which measure would best represent the type of data we have?"

The following table summarizes when each choice of center is most appropriate. It's a good idea verify this using different data sets.

 

Measure Best choice when...
Mean There are no extreme values in the data set
Median

There are extreme values in the data set

There are no big gaps in the middle of the data

Mode The data set has repeated numbers

 

Practice questions

Question 5

The salaries of part-time employees at a company are given in the dot plot below. Which measure of center best reflects the typical wage of a part-time employee?

  1. The mean.

    A

    The mode.

    B

    The median.

    C

    The mean.

    A

    The mode.

    B

    The median.

    C

Question 6

A journalist wanted to report on road speed cameras being used as revenue raisers. She obtained data that showed the number of times $20$20 speed cameras issued a fine to motorists in one month. The results were:

$101,102,115,115,121,124,127,128,130,130,143,143,146,162,162,163,178,183,194,977$101,102,115,115,121,124,127,128,130,130,143,143,146,162,162,163,178,183,194,977

  1. Determine the mean number of times a speed camera issued a fine in that month. Give your answer correct to one decimal place.

  2. Determine the median number of times a speed camera issued a fine in that month. Give your answer correct to one decimal place.

  3. Which measure is most representative of the number of fines issued by each speed camera in one month?

    the mean

    A

    the median

    B

    the mean

    A

    the median

    B
  4. Which score causes the mean to be much greater than the median?

  5. The journalist wants to give the impression that speed cameras are just being used to raise revenue. Which statement should she make?

    A sample of $20$20 speed cameras found that the median number of fines in one month was $136.5$136.5.

    A

    A sample of $20$20 speed cameras found that, on average, $182.2$182.2 fines were issued in one month.

    B

    A sample of $20$20 speed cameras found that the median number of fines in one month was $136.5$136.5.

    A

    A sample of $20$20 speed cameras found that, on average, $182.2$182.2 fines were issued in one month.

    B

Outcomes

MGSE6.SP.5

Summarize numerical data sets in relation to their context, such as by:

MGSE6.SP.5c

Giving quantitative measures of center (median and/or mean) and variability (interquartile range).

MGSE6.SP.5d

Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered.

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