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Mathematical

What is an Outlier Defined as A Level Maths?

“What is an outlier defined as A Level maths?” You may get confused seeing the question as we’ve directly copy pasted the Google search term here. Let’s rephrase it so you’ll understand it a bit better. “What is an outlier defined as in A Level maths” or “Describe outlier meaning in statistics.” Well, as the latter question clearly spits it out, it’s a concept related to statistics. An outlier is a data point in a distribution that breaks away from the overall pattern.

Since you’ve searched the term, we’re going to assume that you’re familiar with statistics and quantitative analysis. So we’re not going to talk about these here today. Instead, we’re going to delve right into your question.

What is an Outlier Defined as A Level Maths?

What are outliers? Instead of going to the definition of outlier in statistics, it’ll be much easier to understand if we describe it.

At A-level, you’ll find the concept of outliers relatively early in your curriculum. Outliers will come up as soon as you come in touch with the following concepts:

To understand an outlier, you first have to be familiar with the following terms:

1. Standard deviations
2. Interquartile range

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Standard Deviations

A standard deviation is a measure of how widely distributed the data is in reference to the mean. A low standard deviation suggests that data is grouped around the mean, whereas a large standard deviation shows that data is more spread out.

In statistics, it is often denoted with the Greek word sigma.

σ

Interquartile Range

The interquartile range in descriptive statistics describes the spread of your distribution’s middle half.

But, What is quartile? Quartiles are values that divide data into four equal parts, similar to how the median divides data into two.

In fact, the median is the second quartile. The second and third quartiles, or the centre half of your data set, are represented by the interquartile range (IQR).

In other words, the difference between the 25th and 75th percentiles is known as the interquartile range. So, the 25th and 75th percentiles are also called the first and third quartiles. As a result, the interquartile range describes the middle 50% of observations.

If the interquartile range is broad, it suggests that the middle 50% of observations are widely apart. The main advantage of the interquartile range is that it can be used as a measure of dispersion/variability even if the extreme values are not captured precisely.

Another advantage is that it is unaffected by extreme values.

Outliers

So, outliers are data points that lie 1.5 times below the 1st quartile or 1.5 times above the 3rd quartile. We can also say that outliers are data points that lie 1.5 times above or below the interquartile range.

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How to Calculate an Outlier?

Outliers can be dealt with using the interquartile range. Because the interquartile range is the middle half of the data, it is reasonable to define an outlier as a specific multiple (1.5 times to be exact) of the interquartile range below the first quartile or above the third quartile.

An exam question might provide you a data set and ask you to compute the interquartile range and identify outliers.

For example, consider the following data set. Imagine that they are annual savings of 12 people in quantities of thousands. So you will also have to interpret the output data as thousands.

 Data Points 1 4 5 5 6 6 6 6 7 10 12 56

Analysis of the 12 Data Points

There are 12 data points here. Let’s analyse them.

 μ or x̅ Mean 10.33333333 Median (0 percentile) 6 σ Standard Deviation 14.64323197 Q1 1st Quartile (25th percentile) 5 Q2 2nd Quartile (50th percentile) 6 Q3 3rd Quartile (75th percentile) 9.25 IQR Interquartile Range Q3 – Q1

The mean here is the average. The median is the middle point of the data set, also called the 50th percentile. The standard deviation is 14.64323197, and it shows how far spread out the data set is from the mean. The interquartile range here is the value when we calculate Q3-Q1.

However, if we look at the data set and exclude 56 out of it, the last data point would be 12, approximately 2.64 less than the standard deviation. It makes very little sense. This happened here because of the data point 56. If we exclude it and recalculate the standard deviation for the remaining 11 data points, it would be 2.891995222, which makes perfect sense.

So you can see how data points that are too far from the mean can affect the statistical analysis. Both mean and standard deviations are highly sensitive to extreme data points.

Statisticians call these extreme data points outliers, which can go both ways- either above the mean, like our example above, or below the mean.

This is also where the interquartile range comes in. It helps us understand where the majority of the data set is and identify outliers.

Using Interquartile Range to Determine Outliers

So, the outlier formula or outlier equation is:

 Outliers below the Mean Q1 – 1.5 x IQR Outliers above the Mean Q3 + 1.5 x IQR

First, calculate the interquartile range and multiply it by 1.5. Then, subtract this value from the 1st quartile and then also add it to the 3rd quartile. The two values that you end up with are the acceptable statistical data range. Any data point outside this would be an outlier.

But there is another way to identify outliers that is common in A level

Using Standard Deviation to Determine Outliers

The following are the formula to identify outliers using the standard deviation.

 Outliers below the Mean x̅ – kσ Outliers above the Mean x̅ + kσ

FAQs

How do you determine an outlier in A level?

An outlier in A level can be determined by looking for data points that are significantly different from the rest of the data set.

How do you use standard deviation to determine outliers in A level maths?

The formula to find outliers using the standard deviation is as follows.

 Outliers below the Mean x̅ – kσ Outliers above the Mean x̅ + kσ

Is an outlier 2 standard deviations from the mean?

Firstly, the mean (or average) of the data set needs to be calculated, followed by the standard deviation, which is a measure of how much the data points vary from the mean. Once the mean and standard deviation have been calculated, any data point that lies outside of 1 or 2 standard deviations away from the mean can be identified as an outlier.

Why do we use 1.5 IQR for outliers?

This number clearly affects the sensitivity of a data set and, thus, the decision rule. Further, if we increase the scale from 1.5 to something greater, some outliers will be included in the data range, severely affecting it.

What is an outlier in math on a box plot?

An outlier is a numerically distant observation from the rest of the data. When examining a box plot, an outlier is defined as a data point that lies outside the box plot’s whiskers. The following is the box plot for our example data set in the blog.

Can you see a little blue dot in the 50-60 range of the Y-axis? That’s the value 56, our outlier.

What is singled out meaning in statistics?

In language, “single out” has a different meaning than it does in statistics. In statistics, we “single out” the outliers and leave them out of the calculation so that they can’t distort the representation of the overall dataset.

Conclusion

So, what is an outlier defined as A Level maths? Extreme data points which can affect the statistical analysis of some given data are outliers. They can cause serious problems if we don’t identify them before we move to draw conclusions on a data set.

The blog has all the instructions to identify outliers. You can also see the example that we provided to get a practical idea of how outliers affect a set of data and how you can easily recognise them.