# Lesson 17 – We who deviate from the norm

Hello, I am the Wood Duck. I am the most pretty of all water birds. I live in wooden swamps; I nest in tree holes or nest boxes along the lake. Don’t mess with me; I have a strong claw, can grip barks and perch on branches.

Ornithologists look for me each year, and migratory bird data center records my detections. These are the numbers they found for the last decade in New York State. Since you are familiar with the number line, I am showing the data as a dot plot to spare you some time. Can you see that I also show the average number of individuals (of my species) on the plot?

Hello, I am the Pine Warbler. I fly in the high pines. I know it is hard to find me, but hey, can’t you listen to my musical trills? I have a stout beak; I can prod clumps of needles. Here are my detections in the last decade in New York State.

Did you notice that my numbers are spread out on the line. Sometimes, my numbers are close to the average; sometimes they are far away from the average. I am deviating from the norm.

😉 These poor ornithologists have to understand the variation in finding me.

🙁 I am not deviating much from the average. You see, the average number of pine warblers in this data is 65, but the maximum is 75 and the minimum is 53. I am close to the average. I don’t deviate from the norm.

Like the average that summarizes the center of the data, if you want to summarize the deviation, you can do the following:

Compute the deviation of each number from the average.

Square these deviations. Why?

Get the average of all these squared deviations.

This measure of the variation in the data is called variance. It is the average squared deviation from the center of the data. Do you think it is sensitive to the outliers?

Mr. Wood, you forgot to point out that if you take the square root of the variance measure, you get the standard deviation.

It is in the same units as the average. We can look at a point on the number line and see how many standard deviations away it is from the average.

In your case, you have a number 79 on the line. Since your standard deviation is around 17, the 79 point is more than two standard deviations away from your average of 42.

Mr. Warbler, thank you. I think our readers now have two measures to summarize the data. The average and the variance. The average provides a measure of the center of the data, and the variance or standard deviation provides a measure of how spread out the data points are from the center.

While they play around with these summary statistics, I will go back to my swimming, and I guess you can go back to your singing.

How many of you thought you would see data on birds here? That is the variance for you. In the data analysis classroom, we not only mean well but also provide variety.

Our friends from the sky have deviated from their norm to teach us standard deviation. Have you been deviating from the norm? If not, why not?

Were you given the index card too?

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