Outliers For the purposes of constructing modified boxplots as...

Question

Outliers For the purposes of constructing modified boxplots as described in Section 3-3, outliers are defined as data values that are above Q3 by an amount greater than 1.5 x IQR or below Q1 by an amount greater than 1.5 x IQR, where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier.

Accepted Answer

Hello. In this video, we are told that a data set is considered to have outliers if any X-value satisfies X less than the first quarter minus 1.5 times IQR, or X greater than the third quarter, plus 1.5 times IQR. This is where Q1 is the first quartile and Q3 is the third quartile, and IQR is the interquartile range, defined as quarter 3 minus quarter 1. For a standard normal distribution, what is the probability that a randomly selected value X is an outlier by this definition? OK, so there's a few things that we need to understand. First, we are assuming that we have a normal distribution. Now, quartile 3, or Q3 represents the third quartile of the normal distribution, and that represents the 75th percentile. The same similarly, Q1 is the first quartile of the standard distribution, and that is going to be the 25th percentile. Now, what we want to do is we first want to find the probability that Z is less than Q1. So if it wants to be an outlier outside of the 25th percentile, it needs to be less than Q1, and this is going to equal to 0.25 since we're talking about the 25th percentile. So, what we need to do is we need to find this value of Q. Now, in order to find the value of Q1, we need to first find the probabilities around 0.25. The probability that Z is less than 0.67 is equal to 0.2514, and the probability that Z is less than -0.68 gives us the value of 0.2483, which has the quartile between this range. And so what we want to do in order to find the first quartile, is we want to take the first quartile Q1. And subtract the probability. Of negative 0.67. Then divide this by negative 0.67, so the probability, again of -0.67 minus the probability of -0.68. That is going to give us 0.25 minus 0.2483, divided by 0.2514 minus. 0.2483. And this is going to allow us to estimate that the first quartile is approximately equal to -0.6745. Now what we want to do is you want to go ahead and do the same thing for quartile 3, which is the 75th percentile. But again, because we have a standard normal distribution centered around 0, we can assume that the 7 75th percentile is equivalent to the negative 25th percentile, and this is going to give us an approximate value of negative. Sorry, positive, 0.6745. So, we have the third quartile and the first quartile, and now what we want to do is we want to find the interquartile range IQR. Now, again, IQR is the third quartile minus the first quartile. So that is going to be 0.6745. Minus negative 0.6745, and this is going to simplify to give us an interquartile range of 1.349. Now what we want to do is we want to calculate the bounds of the interquartile range. Now, the lower bound. Was defined as Q1 minus 1.5 multiplied by IQR. This is going to be 0.6745 Negative 0.6745 minus 1.5, multiplied by RQR which is 1.349, and this is going to simplify. To leave us with a lower bound of negative 2.6977. Now we need to calculate our upper. Our upper limit, and that is going to be Q3 plus 1.5 multiplied by our interquartile range, and that is going to be 0.6745 plus 1.5 multiplied by 1.349, and this will simplify to be 2.6977. Now, for X to be an outlier, In order for X to be an outlier, we need to find the probability that X is less than -2.6977, and add this to the probability that X is greater than 2.9677. This is going to simplify to be 0.0035 plus 1 minus 0.9965, which is going to simplify to be 0.007. So, this is going to be the solution to the problem, and the solution is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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Key Concepts

Outliers

Interquartile Range (IQR)

Normal Distribution

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