Outliers & Cleaning Data | AQA A Level Maths: Statistics Revision Notes 2018

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Outliers

What are outliers?

Outliers are extreme data values that do not fit with the general pattern of the data
They can come from one or two extreme events or from mistakes in the data collection
Outliers will affect some statistics that are calculated from the data
- They can have a big effect on the mean, but not on the median or usually the mode
- The range will be completely changed by a single outlier, but the interquartile range will not be affected
- When calculating the mean or the range it is important to decide whether the outlier(s) should be included in the calculations
  - The question will tell you whether to include the outliers or not
  - You may have to decide which value is the outlier to be removed
  - In general outliers are included if they are a valid piece of data and excluded if it is likely that they are erroneous

How are outliers calculated?

Most of the time within this syllabus the outliers will be a particular distance either side of the interquartile range
- The most common way to calculate an outlier will be using the formulae:
  - A value that is less than $Q_{1} - k$ (interquartile range)
  - A value that is greater than $Q_{3} + k$ (interquartile range)
  - k is a constant that will be given to you in the exam, commonly k=1.5
Outliers could also be situated a number of standard deviations away from the mean
- The most common way to calculate an outlier will be using the formulae
  - A value that is less than $\bar{x} - k σ$
  - A value that is greater than $\bar{x} + k σ$
  - k is a constant that will be given to you in the exam, commonly $k = 2$

How are outliers represented on box plots?

On a box plot an outlier is represented as a cross either side of the maximum or minimum value
If the maximum or minimum value is discovered to be an outlier, the new maximum or minimum value will need to be found for the box plot
- If the data value just above the minimum or just below the maximum is known, this will become the new value
- If the data value is not known, the new minimum or maximum will become the outlier boundary

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Cleaning Data

When should data be cleaned?

The cause of the outlier should be examined by looking into the context of the data
For example:
- a test score of over 100% would most likely be a data collection error
- a single salary that is much higher than the others would likely be for the CEO of the company
If an outlier is determined to be from an error in data collection it should be removed from the data.
- Removing the incorrect data value(s) is called cleaning the data
- It is important to consider very carefully whether you should remove the data value or not
  - If the data value is not an error it should not be removed from the data
If a data value is removed from the data set before calculations are carried out, a justification for the removal of the outlier must be made
Cleaning data also involves removing missing data and errors

Worked example

The ages, in years, of a number of children attending a birthday party are given below:

2, 7, 5, 4, 8, 4, 6, 5, 5, 29, 2, 5, 13,

An outlier is defined as an observation that falls more than $1.5 \times$ the interquartile range above the upper quartile or below the lower quartile

(i)

Identify any outliers within the data set.

(ii)

Clean the data by deciding which values should be removed, justify your answer.

2-3-1-outliers-we-solution

Examiner Tip

Read the question carefully to determine which type of outlier you should be finding and to make sure you are using the correct method.

Outliers & Cleaning Data (AQA A Level Maths: Statistics)

Revision Note

Outliers

What are outliers?

How are outliers calculated?

How are outliers represented on box plots?

Cleaning Data

When should data be cleaned?

Worked example

Examiner Tip

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Author: Amber