## How To Calculate Standard Errors

Business, government and academic activities almost always require the collection and analysis of data. One of the ways to represent numerical data is through graphs, histograms and charts. These visualization techniques allow people to gain better insight into problems and devise solutions. Gaps, clusters and outliers are characteristics of data sets that influence mathematical analysis and are readily visible on visual representations.

## Use Of Domain Knowledge

Sometimes, the typical ranges of a value are known. For example, when measuring blood pressure, your doctor likely has a good idea of what is considered to be within the normal blood pressure range. If they were looking at the values above, they would identify that all of the values that are highlighted orange indicate high blood pressure. As a result, they may advise some course of action.

In this case, âoutliersâ, or important variations are defined by existing knowledge that establishes the normal range. It might be the case that you know the ranges that you are expecting from your data. If you identify points that fall outside this range, these may be worth additional investigation.

## Putting It All Together

Gaps, outliers and clusters in data sets can impact the results of mathematical analysis. Gaps and clusters might represent errors in the data collection methodology. For example, if a telephone survey polls only certain area codes, such as low-income housing complexes or high-end suburban residential areas, and not a broad cross-section of the population, chances are there will be gaps and clusters in the data. Outliers can skew the mean or average value of a data set. For example, the mean or average value of a data set consisting of four numbers — 50, 55, 65 and 90 — is 65. Without the outlier 90, however, the mean is about 57.

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## How To Handle Outliers

If youre worried that an outlier is present in your dataset, you have a few options:

**Make sure the outlier is not the result of a data entry error.**Sometimes an individual simply enters the wrong data value when recording data. If an outlier is present, first verify that the value was entered correctly and that it wasnt an error.**Assign a new value to the outlier**. If the outlier turns out to be a result of a data entry error, you may decide to assign a new value to it such as the mean or the median of the dataset.**Remove the outlier.**If the value is a true outlier, you may choose to remove it if it will have a significant impact on your overall analysis. Just make sure to mention in your final report or analysis that you removed an outlier.

## Outliers: Finding Them In Data Formula Examples Easy Steps And Video

Outliers are stragglers **extremely high or extremely low values** in a data set that can throw off your stats. For example, if you were measuring childrens nose length, your average value might be thrown off if Pinocchio was in the class.

**Contents :**

Watch the video for the definition and how to find outliers with the IQR and Tukeys method:

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## How Can An Outlier Help You Make Good Decision

When an outlier increases or decreases the mean too much, you might decide that it is better to use the median to find the average.

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## What Is An Outlier Definition And Examples

An outlier is a data value that is much greater or much less than the other data values.

Outliers can affect the mean of a group of data and how you interpret your data.

**Example #1:**

On a line plot, an outlier is a data value that is usually located some distance away from other data values.

In the line plot below, 10 is an outlier.

10 is much greater than the other values and looking at the line plot, it is located some distance away from the other values.

How much does 10 affect the mean?

Find the mean with the outlier.

2 × 4 + 6 × 5 + 4 × 6 + 7 × 7 + 10 = 8 + 30 + 24 + 49 + 10 = 121

There are 20 values, so the mean is 121 / 20 = 6.05

Find the mean without the outlier.

2 × 4 + 6 × 5 + 4 × 6 + 7 × 7 + 10 = 8 + 30 + 24 + 49 = 111

There are 19 values now when we remove 10, so the mean is 111/19 = 5.84

The outlier raises the mean by 6.05 – 5.84 = 0.21.

**Example #2:**

In the line plot below, 16 is an outlier.

16 is much smaller than the other values and looking at the line plot, it is located some distance away from the other values.

How much does 16 affect the mean? Find the mean with the outlier.

16 + 3 × 21 + 2 × 22 + 4 × 23 = 16 + 63 + 44 + 92 = 215

There are 10 values, so the mean is 215 / 10 = 21.5

Find the mean without the outlier.

3 × 21 + 2 × 22 + 4 × 23 = 63 + 44 + 92 = 199

There are 9 values now when we remove 16, so the mean is 199/9 = 22.11

The outlier decreased the mean by 22.11 – 21.5 = 0.61.

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## Reasons For Identifying Outliers

We always need to be on the lookout for outliers. Sometimes they are caused by an error. Other times outliers indicate the presence of a previously unknown phenomenon. Another reason that we need to be diligent about checking for outliers is because of all the descriptive statistics that are sensitive to outliers. The mean, standard deviation and correlation coefficient for paired data are just a few of these types of statistics.

## What Is An Outlier In Math

**An outlier is a number in a data set that is much smaller or larger than the other numbers in the data set.****outlier****90****outlier****Determine how the outlier affects the median and the mean in the****following scores.****Determine how the outlier affects the median by finding the median without the outlier included.**

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## What Are Outliers In The Data

*Definition of outliers**outlier**Ways to describe data*

*Box plot construction**Box plots with fences*

## What Is An Outlier

An outlier is a piece of data that is an abnormal distance from other points. In other words, its data that lies **outside the other values** in the set. If you had Pinocchio in a class of children, the length of his nose compared to the other children would be an outlier.In this set of random numbers, 1 and 201 are outliers:1, 99, 100, 101, 103, 109, 110, 2011 is an extremely low value and 201 is an extremely high value.

Outliers arent always that obvious. Lets say you received the following paychecks last month:$225, $250, $25, $235.Your average paycheck is $135. But that small paycheck might be because you went on vacation, so a weekly paycheck average of $135 isnt a true reflection of how much you earned. Your average is actually closer to $237 if you take the outlier out of the set.

Of course, trying to find outliers isnt always that simple. Your data set may look like this:61, 10, 32, 19, 22, 29, 36, 14, 49, 3.You could take a guess that 3 might be an outlier and perhaps 61. But youd be wrong: 61 is the only outlier in this data set.A **box and whiskers chart** often shows outliers:

The outlier on this boxplot is outside of the box and whiskers.

However, you may not have access to a box and whiskers chart. And even if you do, some boxplots may not show outliers. For example, this chart has whiskers that reach out to *include* outliers:

**Need help** with a homework question? Check out our tutoring page!

IQR, Q1 and Q3 found using the online calculator .

*Thats it!*

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## What Does The Skewness Value Tell Us

Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail.

## Measures Of Center Outliers And Averages

MoreVisibility – May25, 2010

Let me take you back to the days when you were an under-21 college student, figuring out who you were and what you wanted to be when you finally grew up. For some of you this may be a lifetime ago, and for others, it may have seemed as if those days happened yesterday .

Most college students must take one, if not two courses in mathematics during their college careers, regardless of their degree program. Most of the time, elementary statistics is the course selected, probably because its the easiest math elective to take for most people. In short, lots of people have an elementary knowledge of statistics. So, why are average-oriented metrics put on such a pedestal?

In elementary statistics, you most likely learned about the four measures of center and about outliers. If you dont remember, thats OK, its probably been a long time since, or you probably werent a math person and wanted to forget everything you had learned as quickly as possible.

The four measures of center are mean, median, mode, and midrange.

**Mean **The mean is what you know as the average. It is calculated by taking all of the values in a set and dividing them by the total number of values in that set. The mean is very sensitive to outliers .

Example: The mean of 1, 3, 5, 5, 5, 7, and 29 is about 7.8571.

Example: The median of 1, 3, 5, 5, 5, 7, and 29 is 5 .

Example: The mode of 1, 3, 5, 5, 5, 7, and 29 is 5 .

Example: The midrange of 1, 3, 5, 5, 5, 7, and 29 is 15 .

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## What Does Outlier Mean In Math

**Asked by: Emmie Moore**

An outlier is **a number that is at least 2 standard deviations away from the mean**. For example, in the set, 1,1,1,1,1,1,1,7, 7 would be the outlier.

**A value that “lies outside” most of the other values in a set of data****compute a z-score for every xi****an extreme value in a data set****average giraffe is 4.8 meters tall****45 related questions found**

## How Do Outliers Affect The Mean

In statistics, the **mean **of a dataset is the average value. Its useful to know because it gives us an idea of where the center of the dataset is located. It is calculated using the simple formula:

**mean** = /

For example, suppose we have the following dataset:

The mean of the dataset is / = **4.6**

But while the mean is a useful and easy to calculate, it does have one drawback: **It can be affected by outliers**. In particular, the smaller the dataset, the more that an outlier could affect the mean.

To illustrate this, consider the following classic example:

Ten men are sitting in a bar. The average income of the ten men is $50,000. Suddenly one man walks out and Bill Gates walks in. Now the average income of the ten men in the bar is $40 million.

This example shows how one outlier could drastically affect the mean.

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## Sample Size & Outliers

The smaller the sample size of the dataset, the more an outlier has the potential to affect the mean.

For example, suppose we have a dataset of 100 exam scores where all of the students scored at least a 90 or higher except for one student who scored a zero:

The mean turns out to be **93.18**. If we removed the 0 from the dataset, the mean would be **94.12**. This is a relatively small difference. This shows that even an extreme outlier only has a small effect if the dataset is large enough.

## Provide Confidence In Analysis

When outliers exist in our data, it can affect the typical measures that we use to describe it.

For example, if we had five friends with the ages of 23, 25, 27, and 30, the average age would be 26.25.

In this case we can have high confidence that the average of our data is a good representation of the age of a âtypicalâ friend.

However, if we then change the value final value and we had friends with the ages of 23, 25, 27, and 70, the average age is now 36.25. This is quite a large increase, even though the majority of our friends are under 30 .

In this case, we have much less confidence that the average is a good representation of a typical friend and we may need to do something about this.

Being able to identify outliers can help to determine what is typical within the data and what are exceptions. If we donât have outliers, this can increase our confidence in the consistency of our findings.

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## Examples Of Outlier In A Sentence

*outlier**Quartz**outlier **Travel + Leisure**outlier **The New Republic**outlier **The Indianapolis Star**outlier **cleveland**outlier **Vulture**outlier **San Francisco Chronicle**outlier **WSJ*

These example sentences are selected automatically from various online news sources to reflect current usage of the word ‘outlier.’ Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback.

## How To Find Outliers With The The Tukey Method

Frequency chart with boxplot at the top. The outliers are shown as dots outside the range of the whiskers.

The Tukey method for finding outliers uses the interquartile range to filter out very large or very small numbers. Its practically the same as the procedure above, but you might see the formulas written *slightly* differently and the terminology is a little different as well. For example, the Tukey method uses the concept of fences.

The formulas are:Low outliers = Q1 1.5 = Q1 1.5High outliers = Q3 + 1.5 = Q3 + 1.5Where:Q3 = third quartileIQR = Interquartile range

These equations give you two values, or **fences**. You can think of them as a fence that cordons off the outliers from all of the values that are contained in the bulk of the data.

**Sample question:** Use Tukeys method to find outliers for the following set of data: 1,2,5,6,7,9,12,15,18,19,38.Step 1: Find the Interquartile range:

Step 2: Calculate 1.5 * IQR:1.5 * IQR = 1.5 * 13 = 19.5

Step 3: Subtract from Q1 to get your lower fence:5 19.5 = -14.5

Step 4: Add to Q3 to get your upper fence:18 + 19.5 = 37.5.

*Thats how to find outliers with the Tukey method!*

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## Small & Large Outliers

An outlier can affect the mean by being unusually small or unusually large. In the previous example, Bill Gates had an unusually large income, which caused the mean to be misleading.

However, an unusually small value can also affect the mean. To illustrate this, consider the following example:

Ten students take an exam and receive the following scores:

The mean score is

84.6.However, if we remove the 0 score from the dataset, then the mean score becomes

94.

The one unusually low score of one student drags the mean down for the entire dataset.

## Modified Thompson Tau Test

The modified Thompson Tau test is a method used to determine if an outlier exists in a data set. The strength of this method lies in the fact that it takes into account a data set’s standard deviation, average and provides a statistically determined rejection zone thus providing an objective method to determine if a data point is an outlier.How it works:First, a data set’s average is determined. Next the absolute deviation between each data point and the average are determined. Thirdly, a rejection region is determined using the formula:

- Rejection Region

| .If > Rejection Region, the data point is an outlier.If Rejection Region, the data point is not an outlier.

The modified Thompson Tau test is used to find one outlier at a time . Meaning, if a data point is found to be an outlier, it is removed from the data set and the test is applied again with a new average and rejection region. This process is continued until no outliers remain in a data set.

Some work has also examined outliers for nominal data. In the context of a set of examples in a data set, instance hardness measures the probability that an instance will be misclassified ( 1

trained on training set t with hyperparameters . Instance hardness provides a continuous value for determining if an instance is an outlier instance.

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