什么是离群值？

So far in statistics, we have learned about the range, the interquartile range (IQR), and how we can use both of these to compare data with the box and whisker plot. However, the interquartile range also has other uses: it can help us find outliers!

##Outliers

Outliers are data points in a data set that are too high or low compared to the rest of the dataset. When conducting an experiment, outliers are often a result of experimental error. If an extreme data point can be proven to be an outlier, then researchers will often exclude the data point from their analysis. But, we need to prove whether a suspicious data point is truly an outlier!

##Determining Outliers

Lucky us! There are a few very handy formulas we can use to find out if a datapoint is an outlier:

Lower quartile $- (1.5 \times IQR)$.

Any data point below this value is an outlier

Upper quartile $+ (1.5 \times IQR)$.

Any data point above this value is an outlier

##Let's work through an example!

We have a dataset of numbers: $8, 13, 1, 16, 5, 7, 9, 34$

Just looking at these numbers, there are a few that stand out as being too extremely large or small: $1$ and $34$. However, we cannot just say these are outliers, we have to prove it. Let's use the formulas.

First, we have to calculate the interquartile range. Remember, the IQR is the difference between the upper (third) quartile and the lower (first) quartile (Q1 and Q3).

To find the upper quartile (Q3), we have to find the median of the upper half of the dataset. To find the lower quartile (Q1), we have to find the median of the lower half of the dataset.

Hint! It helps to reorder the dataset in numerical order:

Dataset: $1, 5, 7, 8, 9, 13, 16, 34$

In this dataset, the upper quartile is $14.5$. The lower quartile is $6$.

Therefore, the IQR $= 8.5$.

Now let's use the first formula to find the lower limit for any outliers.

Remember, the formula is: lower quartile $- (1.5 \times IQR)$. $$6 - 1.5(8.5) = -6.75$$

Since $1$ is greater than $-6$, we know that 1 is not an outlier. This is a great example of why we need the outlier test because even though $1$ seems much smaller than the other numbers, it's not actually an outlier.

Let's try the second formula to find the upper limit for any outliers. Remember, the formula is: Upper quartile $+ (1.5 \times IQR)$.

$$14.5 + 1.5(8.5) = 27.25$$

Because $34$ is greater than $27.25$, we know that $34$ is an outlier.

(Video) Statistics - How to find outliers

by MySecretMathTutor

So far, we have learned about, variability, range, and quartiles, but now let's add another element by learning about outliers.

Watch this video by MySecretMathTutor to learn how to find and use outliers.

An outlier is a very high or low value in a data set. It can skew a dataset, so often statisticians decide to remove an outlier from a dataset.

The formula for finding an outlier is any number greater than the upper quartile $+ (1.5\times IQR)$ or lower than the lower quartile $-(1.5 \times IQR)$.

Remember, the IQR refers to the interquartile range and the upper and lower quartiles are mentioned in the lesson on box and whisker plots.

Try out the example problem in the video to see if there are any outliers in the dataset.

Outliers

When dealing with large real-life datasets, we will often come across outliers.

An outlier is a point in the dataset that lies outside of the typical range of values for that dataset, far from the median of the dataset. In simplified terms, you can think of an outlier as a point that is extremely high or extremely low, relative to the rest of the dataset.

There is a formula for finding outliers. Identifying outliers is not a guessing game, thankfully!