1.04 Mode, median and mean

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统计基础

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University of Amsterdam

统计基础

1926 個評分

課程 3（共 5 門，Specialization 社会科学的方法和统计）

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.

從本節課中

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.

1.04 Mode, median and mean6:57

1.05 Range, interquartile range and box plot7:55

1.06 Variance and standard deviation5:14

與講師見面

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

Next to summarizing a distribution by means of graphs.

It can also be useful to describe the center of your distribution.

There are three main ways in which you can do that.

By means of the mode, the median and by means of the mean.

These three m's are often referred to as measures of central tendency.

Finding the mode is easy, it is the value that occurs most frequently.

It is, in other words, the most common outcome.

The mode is often used as a measure of central tendency

if a variable is measured on a nominal or ordinal level.

In this pie chart, you can see which continent

players in the main Spanish football competition come from.

The pie chart makes immediately clear what the mode is, it is Europe.

70% of the players was born in Europe.

Know that the mode here is Europe,

which is the name of the category that appears most often.

The mode is not 70%,

that's just the percentage of observations that fall in that specific category.

You can also have more than one mode.

Imagine that there exists a football player that strongly

divides football fans.

Some people find him very sympathetic,

while others find him strongly unsympathetic.

Let's name this player, Franco Galton.

Imagine you have asked the representative sample of the Spanish population of 500

respondents, what they think of Franco Galton?

Your respondents could indicate on a scale from 0 to 10,

how sympathetic they think he is.

0 refers to very unsympathetic, and 10 refers to very sympathetic.

Let's say that this is the shape of the histogram resulting from this study.

You can see that the Spanish population is strongly divided.

Some find Galton very unsympathetic, and some find him very sympathetic.

As you can see the distribution has two modes, 3 and 8.

This is clearly a bi-modal distribution.

The second measure of central tendency is the median.

The median is nothing more than the middle value of your observations when they

are ordered from the smallest to the largest.

Imagine you have also asked seven of your respondents what they think

of another famous football player named Tomas Bayez.

Let's assume that this is the data matrix of his study.

The mode here is 8, the value that occurs most often.

To compute a median, we first have to order all values from low to high.

This is the result.

Then we have to pick the middle value.

So, the median is 8.

It is slightly more complicated if we have an even number of cases instead of an odd

number of cases.

Imagine we haven't asked 7, but 8 people what they think of Tomas Bayez.

This is the data matrix.

And this is the order of the values from low to high.

However, in this case, there is no single middle value.

How do we solve that problem?

Well, we just take the average of the two middle values.

That 7 and 8, dived by 2 equals 7.5.

The median in this case is 7.5.

Notice that the median divides the distribution into two equal parts.

50% of the values lies below the median.

And 50% above the median.

The third measure of central tendency, is the most often used one and also the one,

you most probably already know quite well. It's the mean.

The mean is the sum of all the values divided by the number of observations.

This is the formula with which you can compute the mean,

it looks more complicated than it is.

The formula tells you that the mean of variable x symbolized as x bar, equals

the sum of all the values of x divided by the sample size, which is symbolized by n.

To give an example, let's again use the study on Tomas Bayez.

This was the data matrix.

The formula tells us to first sum all the values.

That's 6 plus 7, plus 7, plus 8, plus 8, plus 8, plus 9.

That equals 53.

We now have resolved this part of the formula.

We now have to divide by n.

The sample size in this study is 7, so 53 divided by 7 = 7.6.

The mean is 7.6.

You can think of the mean as the balance point of your data.

Imagine we would place weights on a balance.

One for each observation.

Then the mean is the point on the balance where the total weight on the one side

exactly equals the weight on the other side.

You're quite familiar now with the three m's.

And you can easily compute the middle of a group of scores in various ways.

But when should you report which measure of central tendency?

That partially depends on the the measurement level of your variable.

If it's nominal, it is impossible to compute the median, or the mean.

Think about it, you cannot apply numerical operations on nominal variables, nor

can you order them.

The only appropriate measure of central tendency,

when a variable is nominal, is the mode.

But what to do in case of a quantitative variable?

Imagine you're sitting in a canteen of a football club in your hometown and

you would like to compute the mean and median income of all persons present.

That's you, 5 other guests and the bartender.

This is the data matrix, the mean is around 35,000.

The median is exactly 35,000.

Their pretty close to each other, and

it doesn't matter which one you use to describe the center of your distribution.

But now image the famous football player Franco Galton walks into the canteen.

Say he gets about 70 million per year, the median increases slightly to 36,000.

The mean however becomes more than 8 million now.

We say that Franco Galton is an outlier in this distribution.

He earns much more than all the other people present and

his income exerts a disproportional effect on the mean income.

In this case, it might be argued that it makes more sense to compute a median

than the mean to describe the center of the distribution.

Let me briefly summarize what you've learned in this video.

To describe the center of a distribution you can use three measures

of center tendency, the mode, the median, and the mean.

If your variable is categorical, you use the mode, and

if it's quantitative, you employ the median or the mean.

Go for the median if you have influential outliers or

if the distribution is highly skewed, and if that's not the case, go for the mean.

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