1.07 Z-scores

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From the course by University of Amsterdam

Basic Statistics

1712 ratings

University of Amsterdam

Basic Statistics

1712 ratings

Course 3 of 5 in the Specialization Methods and Statistics in Social Sciences

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

From the lesson

Exploring Data

In this first module, we’ll introduce the basic concepts of descriptive statistics. We’ll talk about cases and variables, and we’ll explain how you can order them in a so-called data matrix. We’ll discuss various levels of measurement and we’ll show you how you can present your data by means of tables and graphs. We’ll also introduce measures of central tendency (like mode, median and mean) and dispersion (like range, interquartile range, variance and standard deviation). We’ll not only tell you how to interpret them; we’ll also explain how you can compute them. Finally, we’ll tell you more about z-scores. In this module we’ll only discuss situations in which we analyze one single variable. This is what we call univariate analysis. In the next module we will also introduce studies in which more variables are involved.

1.07 Z-scores4:38

1.08 Example6:46

Meet the Instructors

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

What you see here is the so-called tattoo density of football players expressed in

the percentage of the body covered with tattoos.

The dot plots and the standard deviations show that there is much more variability

in the distribution of team 2 than in the distribution of team 1.

Sometimes researchers ask the question if a specific observation is common or

exceptional.

To answer that question, they express a score in terms of the number of standard

deviations it is removed from the mean.

This number is what we call a z-score.

In this video, I'll explain how you can compute z-scores, and

I'll tell you why they can be useful.

Let's first take a look at the distribution of Team 1.

The mean is 15, and the standard deviation is 2.5.

To compute z-scores, we use this formula.

It is not very complicated.

It tells you to compute for

the value you're interested in, the difference between that value, and

the mean, and to divide the outcome by the standard deviation.

Let's see what it means for a tattoo density of 10.8%.

The z-score of that value is 10.8- 15 divided by 2.5.

That equals -1.68.

So the z-score is -1.68.

You can do that for all the values in your distribution.

If you do that, these are the results.

Notice that you end up with negative z-scores and positive z-scores.

Negative z-scores represent values below the mean, and

positive z-scores represent values above the mean.

Because the mean is the balance point of your distribution, the negative and

positive z-scores cancel each other out.

In other words, if you add up all z-scores, you will get a value of 0.

Okay, that's nice.

But how do you know if a certain z-score is low or high?

Well, that depends on your distribution and on context.

A good rule is that if the histogram of your variable is bell shaped,

68% of the observations fall between z-scores -1 and 1.

95% between z-scores of -2 and 2, and

99% between z-scores of -3 and 3.

This means that for this type of distribution, a z-score of more than 3 or

less than -3 can be conceived of as rather exceptional.

However, if a distribution is strongly skewed to the right, as in this graph,

large positive z-scores are more common,

because there are more extreme values on the right side of the distribution.

Similarly, if a distribution is strongly skewed to the left,

large negative z-scores are more common because there are more

extreme values on the left side of the distribution.

A rule that applies to any distribution regardless shape,

is that 75% of the data must lie within a z-score of plus or minus 2.

And 89% within a z-score of plus or minus 3.

So in itself a z-score gives you, to a certain extent,

information about how extreme an observation is.

Z-scores are even more useful if you want to compare different distributions.

Let's, for example,

look at the question whether a body weight of 19.3 is common or not.

Well, in team 1 it is not that common.

The z-score is 19.3- 15 divided by 2.5.

That equals 1.72.

In team 2, the value of 19.3 corresponds with

a z-score of 19.3- 15 divided by 8 = 0.54.

This indicates that in team 2,

the value of 19.3 is much more common than in team 1.

In team 2, it is only 0.54 standard deviations removed from the mean.

In team 1, it is 1.72 standard deviations removed from the mean.

If we recode original scores into z-scores,

we say that we standardize a variable.

Standardization means that we replace the scores measured in the original metric

by scores expressed in standard deviations from the mean.

The advantage is that we can see at a glance whether a specific score

is relatively common or exceptional.

So, is a football player covering about one-fifth of his

body with tattoos exceptional or not?

Well, that depends on the football team or

whichever other group you want to compare him with.

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