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In some data sets, there are values (observed data points) called outliers . Outliers are observed data points that are far from the least squares line. They have large "errors", where the "error" or residual is the vertical distance from the line to the point.

Outliers need to be examined closely. Sometimes, for some reason or another, they should not be included in the analysis of the data. It is possible that an outlier is a result of erroneous data. Other times, an outlier may hold valuable information about the population under study and should remain included in the data. The key is to examine carefully what causes a data point to be an outlier.

Besides outliers, a sample may contain one or a few points that are called influential points . Influential points are observed data points that are far from the other observed data points in the horizontal direction. These points may have a big effect on the slope of the regression line. To begin to identify an influential point, you can remove it from the data set and see if the slope of the regression line is changed significantly.

Computers and many calculators can be used to identify outliers from the data. Computer output for regression analysis will often identify both outliers and influential points so that you can examine them.

Identifying outliers

We could guess at outliers by looking at a graph of the scatterplot and best fit-line. However, we would like some guideline as to how far away a point needs to be in order to be considered an outlier. As a rough rule of thumb, we can flag any point that is located further than two standard deviations above or below the best-fit line as an outlier . The standard deviation used is the standard deviation of the residuals or errors.

We can do this visually in the scatter plot by drawing an extra pair of lines that are two standard deviations above and below the best-fit line. Any data points that are outside this extra pair of lines are flagged as potential outliers. Or we can do this numerically by calculating each residual and comparing it to twice the standard deviation. On the TI-83, 83+, or 84+, the graphical approach is easier. The graphical procedure is shown first, followed by the numerical calculations. You would generally need to use only one of these methods.

In the third exam/final exam example , you can determine if there is an outlier or not. If there is an outlier, as an exercise, delete it and fit the remaining data to a new line. For this example, the new line ought to fit the remaining data better. This means the SSE should be smaller and the correlation coefficient ought to be closer to 1 or –1.

Graphical identification of outliers

With the TI-83, 83+, 84+ graphing calculators, it is easy to identify the outliers graphically and visually. If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance were equal to 2 s or more, then we would consider the data point to be "too far" from the line of best fit. We need to find and graph the lines that are two standard deviations below and above the regression line. Any points that are outside these two lines are outliers. We will call these lines Y2 and Y3:

As we did with the equation of the regression line and the correlation coefficient, we will use technology to calculate this standard deviation for us. Using the LinRegTTest with this data, scroll down through the output screens to find s = 16.412 .

Line Y2 = –173.5 + 4.83 x –2(16.4) and line Y3 = –173.5 + 4.83 x + 2(16.4)

where ŷ = –173.5 + 4.83 x is the line of best fit. Y2 and Y3 have the same slope as the line of best fit.

Graph the scatterplot with the best fit line in equation Y1, then enter the two extra lines as Y2 and Y3 in the "Y="equation editor and press ZOOM 9. You will find that the only data point that is not between lines Y2 and Y3 is the point x = 65, y = 175. On the calculator screen it is just barely outside these lines. The outlier is the student who had a grade of 65 on the third exam and 175 on the final exam; this point is further than two standard deviations away from the best-fit line.

Sometimes a point is so close to the lines used to flag outliers on the graph that it is difficult to tell if the point is between or outside the lines. On a computer, enlarging the graph may help; on a small calculator screen, zooming in may make the graph clearer. Note that when the graph does not give a clear enough picture, you can use the numerical comparisons to identify outliers.

The scatter plot of exam scores with a line of best fit.Two yellow dashed lines run parallel to the line of best fit. The dashed lines run above and below the best fit line at equal distances. One data point falls outside the boundary created by the dashed lines—it is an outlier.
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Questions & Answers

for poisson distribution mean............variance.
mehul Reply
both are equal to mu
what is a variable
Bonolo Reply
something that changes
why we only calculate 4 moment of mean? asked in papers.
Faizan Reply
why we only 4 moment of mean ? asked in BA exam
Good evening, can you please help me by sharing regression and correlation analysis notes....thank you in advance
Refiloe Reply
Hello, can you please share the possible questions that are likely to be examined under the topic: regression and correlation analysis.
for normal distribution mean is 2 & variance is 4 find mu 4?
Faizan Reply
repeat quastion again
find mu 4. it can be wrong but want to prove how.
for a normal distribution if mu 4 is 12 then find mu 3?
Faizan Reply
Question hi wrong ha
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if X is normally distributed. (n,b). then its mean deviation is?
The answer is zero, because all odd ordered central moments of a normal distribution are Zero.
which question is zero
sorry it is (5,16) in place of (n,b)
I got. thanks. it is zero.
How did we get the 24000
patrick Reply
where do I start
Jeremy Reply
in a large restaurant an average of every 7 customers ask for water with the their meal. A random sample of 12 customers is selected, find the probability that exactly 6 ask for water with their meal
Rufai Reply
any body with idea
conditional probability
can you help please
iam really sorry. it's been long since I used these things. I just gave you a hint though
this follows binomial distribution. p(X=6)=12C6*(0.6)^6*0.4^6 use this formula n find.
can you explain the cosidered variable in the formula
x is variable wich is exactly 6 costumers
n is number of customers
oh right !!! thanks yaar
I agree with Seyda too
I agree with Syeda too
7/12 =0.58is it?
what is descriptive statistic
Shadow Reply
Descriptive statistics are brief descriptive coefficients that summarize a given data set, which can be either a representation of the entire or a sample of a population. Descriptive statistics are broken down into measures of central tendency and measures of variability (spread). 
are you getting this ?
if so let me know
yes m getting
what's taking place can l join u
yeah !!why not? sure
okey thanks
where are statistics used
akinola Reply
how u doing
please guys uses of statistics....
james Reply
the upper quartile of the population 10,12,14,16,18,20,25,15,11,11,17,is................?
The probability range is 0 to 1... but why we take it 0 to 1....
Muhammad Reply
because in probability 1 means success and 0 means failure and it cnnt be more or less than 1 and 0.
b/c v hv mazimum probibliy 1 and minimum which is.no.probiblity is 0.so.v hv the range from 0 to 1
the size of a set is greeter than its subset
The probability of an event will not be less than 0.  This is because 0 is impossible (sure that something will not happen).The probability of an event will not be more than 1. This is because 1 is certain that something will happen
what do they mean in a question when you are asked to find P40 and P88
Megrina Reply
I dont get your question! What are you talk ING about?
you're asked to find page 40 and page 88 on that particular book.
any suggestions for statistics app better than this
sorry miss wrote the question
No problem) By the way. I NEED a program For statistical data analysis. Any suggestion?
Eviews will help u
arey there any data analyst and working on sas statistical model building
Hi guys ,actually I have dicovered that the P40 and P88 means finding the 40th and 88th percentiles 😌..
who can explain the euclidian distance
I am fresh student of statistics (BS) plz guide me best app or best website relative to stat topics
IMAGESNEWSVIDEOS A Dictionary of Computing. measures of location Quantities that represent the average or typical value of a random variable (compare measures of variation). They are either properties of a probability distribution or computed statistics of a sample. Three important measures are the mean, median, and mode.
Ahmed Reply

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