Mba724 s3 w2 central tendency & dispersion (chung)

This lecture presentation complements Khan’s tutorials

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In this lecture we will discuss the different methods to measure central tendency and

dispersion in a statistical sample.

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Central tendency is just a technical way of saying, what’s typical of this sample? For

example, out of all Carlow students, which gender is the more typical one? Male or female?

Out of all the products listed on Amazon, which is the best seller? And out of all the eBay

listings of “Tickle Me Elmo,” which price is the most common one?

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These three different measures are discussed in detail by Khan Academy. Here are some

brief summaries

We will discuss normal distribution.

One key idea is this:

If the sample is normally distributed, meaning it looks like a symmetrical bell curve, then

mean, median and mode will be the same number.

However, if the sample is skewed either to the left or to the right, then these three

numbers would take on different values.

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Concepts like mean and standard deviation are really based on the theory of normal curve

Note it’s a theory, a conceptualization of how data should be distributed in an ideal world

In reality, often times distributions are not perfectly normal

Next slide is an example

Note that the mean is 50 percentile

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Look at this distribution of salary data

It’s heavy on the left side, with a long skinny tail on the right

Definitely not symmetrical

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When we impose the normal curve on top of the salary distribution,

We see that the normal curve only captures the right tail well

For the left tail, the normal doesn’t describe the actual distribution very well

This is because the salary data is positively skewed

In skewed data, mode and median describe the central tendency better than the mean

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In addition to central tendency, we also need a way to describe how spread out the distribution is, and how weird a case is (relative to the mean)

When a case is very close to the mean, we have an average joe.When a case is far off from the mean on the tip of a long tail, we have a weirdo!

In real life, we often discuss dispersion without realizing it. For example:In which percentile is my child’s height?How many people in this class will get an A?Is the customer’s credit score above or below average? By how much?Is a donation of $30,000 pretty common or very rare? How rare is it?

This slide illustrates the distribution of total purchase after a customer clicks on a link.Look at the data, the mean, the distribution, and reflect on the following questions:

How likely would an average customer spend $200 per order?�Very unlikely – it’s at the end of the curve – in a tail

How about $35?���� Much more likely – it’s the average order

In what percentile is a $67 order?�84% - we know because it’s one standard deviation (34%) above the mean (50%)

The next slide explains what’s a standard deviation

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Standard deviation is a standardized measure of dispersion

It tells you whether the distribution is short and fat (with a big standard distribution) or tall

and skinny (with a small standard distribution)

The calculation is explained well by Khan

The basic idea to take away is:

The standard deviation tells you, on average, how far away the data points are from the

mean

For example, let’s say that the Steelers have an average score of 25 per game, and the

standard deviation is 1. Let’s also say that the Greenbay Packers have an average score of

25 per game, and a standard deviation of 7.

In this example, both teams are comparable in terms of average scores, but the Steelers

have a much smaller standard deviation. This means the Steelers’ performance is pretty

consistent over time, their scores may be above or below 25, but only by 1-2 points on

average. If you plot their scores on a chart, you would see that most of them pack around

25, with a nice narrow distribution that peaks around 25.

In contrast, the Packers may average around 25, but their performance varies widely from

game to game. One day they may score 18 (25-7) and the next day they may score 32

(25+7) If you plot their widely varied scores on a chart, you would get a short and fat

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distribution.

(Go Steelers Go!)

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What are practical ways to use the standard deviation?

With a normal distribution, the mean divides it up evenly in the middle. The portion below

the mean covers 50% of the population, whereas the portion above the mean also covers

50% of the population.

The first standard deviation away from the mean covers 34% of the distribution.

In other words, 1 standard deviation above the mean is 50% + 34% which is 84 percentile

Let’s say that the average weight for a one year old is 25 pounds, and a standard deviation

of 2 pounds.

Connor is 23 pounds. That’s 1 standard deviation below the mean. In other words he is

50%-34% or 16th percentile of the population

Nardia is 27 pounds. That’s 1 standard deviation above the mean. In other words she is

50%+34% or 84th percentile of the population

The entire distribution is covered by roughly 6 standard deviations – 3 above the mean and

3 below the mean

Hence the name of the quality management program “Six Sigma”

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More examples:

Given a mean and a standard deviation score, you have a pretty good idea of what the distribution is like – is it fat and short, or tall and skinny?

We can then map out individual scores on the distribution and tell the average joes from the weirdos!

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The Z score is the number of standard deviations fro the meanWith our previous example, Connor would have a Z score of 1, while Nardia has a Z score of negative 1.

The average joes would have close to zero z scores (e.g., 0.0006, -.0029)Whereas the weirdos have extremely large or small z scores (e.g., 3.07, -2.99)

Again -The z score is the number of standard deviations a data point is away from the meanLet's say that the average weight for all American women is 150, and the standard deviation is 20.If your weight is 130, then your z score is -1, because you're exactly 1 standard deviation below the mean.If Peggy's weight is 170, then her z score is 1, because she is exactly 1 standard deviation above the mean.

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Questions? Schedule a chat/phone meeting with the instructor for more assistance

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