How to Identify Outliers Using IQR Formula and Examples
An outlier is a mathematical value in a set of data which is quite distinguishing from the other values. In simple terms, outliers are values uncommonly far from the middle. Mostly, outliers have a significant impact on mean, but not on the median, or mode. Thus, the outliers are crucial in their influence on the mean. Remember that there is no rule to determine the outliers. Value of an outlier is generally more than 1.5 times the value of the interquartile range (IQR) beyond the quartiles.
What is an Outlier in a Line Plot?
Plotting the data on a number line as a dot plot will enable you to determine the outliers.
Outlier Examples
Outliers are basically considered to be stragglers, meaning that — extremely high or extremely low values — in a data that can throw off the stats. For example, if you were measuring the height of people in a room, your average value might be thrown off if Robert Wadlow was in the room.
Apparently, Robert Wadlow is discovered to be the tallest man ever in medical history, who when last measured to be 2.72 m (8 ft 11.1 in) tall on 27 June 1940.
Displaying Outliers in Box and Whisker Plots
Box and whisker plots will often display outliers as dots that are individualized from the rest of the plot.
Below are a box plot and whisker plot of the distribution from above that does not display outliers.
(Image will be uploaded soon)
Below, is a box and whisker plot of a similar distribution that does display outliers.
(Image will be uploaded soon)
Solved Examples
Below is the step-by-step solution to the outlier math example.
Example:
Determine the outliers of the data set. Also, evaluate the mean of the data set including the outliers and excluding the outliers.
35, 75, 20, 25, 15, 30, 30, 15, 45, 40, 110
Solution:
First, arrange the data set in order.
15, 15, 20, 25, 30, 30, 35, 40, 45, 75, 110
Now, plot the data on a number line in the form of a dot plot.
The values 75 and 110 are far off the middle. Thus, these two values are outliers for the assigned set of data.
Find the mean median mode outlier of the data:
Mean = {Sum of the data values}/{Number of data values}
= [15 + 15 + 20 + 25 + 30 + 30 + 35 + 40 + 45 + 75 + 110]/ 11
= 40
Now to find the mean without the outlier,
Evaluating the mean of the data set excluding the outliers, remove the values far off the middle (i.e. 75 and 110):
Mean = Sum of the data values/Number of data values
= {15 + 15 + 20 + 25 + 30 + 30 + 35 + 40 + 45}/9
=20.45
The mean of the given data set is 40 when outliers are included, however, it is 20.45 when outliers are not included.
Example:
For the data set including values 2, 5, 6, 9, 12, we are available with the following five-number summary:
Solution:
Minimum = 2
1st Quartile = 3.5
Median = 6
3rd Quartile = 10.5
Maximum = 12
IQR = 10.5 – 3.5 = 7.
Thus, 1.5·IQR = 10.5.
In order to identify if there are any outliers, we should consider the numbers that are 1.5·IQR or 10.5 beyond the quartiles.
1st quartile – 1.5·IQR = 3.5 – 10.5 = –7
3rd quartile + 1.5·IQR = 10.5 + 10.5 = 21
Considering the fact that none of the data lies outside the interval from –7 to 21, thus, we deduce there are no outliers.
Did You Know?
The outlier is a data point that lies outside the entire pattern in a distribution.
The outliers are shown as dots.
A usual rule says that a data point is an outlier given that it is more than 1.5 IQR1.
The whiskers are required to change.
Whiskers stretch out to the farthest point in the data set that isn't an outlier.
FAQs on Understanding Outliers in Statistics
1. What is an outlier in statistics?
An outlier is a data value that lies unusually far from the rest of the observations in a dataset. It may be much higher or lower than most values and can affect measures like the mean and standard deviation. Outliers can occur due to data entry errors, measurement errors, or genuine extreme values in real-life situations.
2. How do you identify an outlier using the IQR method?
An outlier using the Interquartile Range (IQR) method is any value below Q1 − 1.5×IQR or above Q3 + 1.5×IQR. Follow these steps:
- Find Q1 (first quartile) and Q3 (third quartile).
- Compute IQR = Q3 − Q1.
- Calculate lower bound = Q1 − 1.5×IQR.
- Calculate upper bound = Q3 + 1.5×IQR.
- Any value outside these bounds is an outlier.
3. What is the formula for detecting outliers using the Z-score?
An outlier using the Z-score method is typically a value with a Z-score greater than ±3. The formula is Z = (x − μ) / σ, where μ is the mean and σ is the standard deviation. If |Z| ≥ 3, the data point is considered an extreme value in a normal distribution.
4. How do outliers affect the mean and median?
Outliers significantly affect the mean but have little effect on the median.
- The mean shifts toward the extreme value.
- The median remains relatively stable because it depends on position, not magnitude.
- This is why the median is preferred for skewed data with outliers.
5. Can you give an example of finding an outlier?
Yes, an outlier can be identified using the IQR method in a dataset like 2, 4, 5, 6, 7, 20.
- Q1 = 4, Q3 = 7
- IQR = 7 − 4 = 3
- Upper bound = 7 + 1.5×3 = 11.5
- Lower bound = 4 − 1.5×3 = -0.5
- The value 20 is above 11.5, so it is an outlier.
6. What is the difference between an outlier and an extreme value?
An outlier is a statistically defined unusual value, while an extreme value is simply a very high or low observation.
- Outliers are identified using rules like IQR or Z-score.
- Extreme values may not always meet statistical criteria.
- All outliers are extreme values, but not all extreme values are outliers.
7. Should outliers always be removed from a dataset?
Outliers should not always be removed; they should only be removed if they are errors or irrelevant to the analysis.
- Check for data entry or measurement mistakes.
- If the value is genuine, it may contain important information.
- Removing valid outliers can bias statistical results.
8. How are outliers shown in a box plot?
In a box plot, outliers are displayed as individual points beyond the whiskers.
- The box represents Q1 to Q3.
- The line inside the box shows the median.
- Whiskers extend to 1.5×IQR from the quartiles.
- Points beyond the whiskers are marked as outliers.
9. What causes outliers in data?
Outliers are caused by errors, variability, or rare events in the data.
- Data entry or recording mistakes.
- Measurement or experimental errors.
- Natural variation or genuine extreme cases.
- Sampling from different populations.
10. Why are outliers important in statistical analysis?
Outliers are important because they can influence statistical results and reveal unusual patterns.
- They can distort the mean and standard deviation.
- They may indicate errors that need correction.
- They can highlight rare but meaningful events.
- They affect regression analysis and hypothesis testing.
