In this video I explain the strengths and limitations of the 3 measures of central tendency: the mean, the median, and the mode. This is part of a key theme of statistics, which is the importance of the choices researchers make in how to summarize and present their data. I explain how each measure of central tendency is calculated using examples, describe the use of Greek letters for parameters in comparison to the statistics used for samples, and consider practical situations in which each measure might be more or less revealing of patterns in the data.
Video Transcript
Hi, I’m Michael Corayer and this is Psych Exam Review. In this video I’m going to explain the three measures of central tendency sometimes called measures of central location. Now these are fairly basic statistics but they bring up one of the key themes of statistics, which is that we need to summarize our data. If researchers just gave us all the raw data it would be highly accurate but it would be impractical. We want to know what the data means. We need it to be summarized and presented. You can think of the raw data as a giant block that has all the details and summarizing necessarily chips away at that detail, but hopefully it does so in a way that helps us understand what’s underneath. We want to extract information about the patterns in our data in order to improve our understanding.
Now this means that researchers must make choices and it’s important that they understand the implications of those choices when going from raw data to summaries, analyses, or visualizations. This is true for these basic statistics and it’s also true for the more advanced concepts that we’ll see later. Any measure of central tendency tells us about the average of our data but this term average is often used to refer to the mean so we might be better off just thinking about the center of our data or about what’s typical in our responses.
The mean or the arithmetic mean is calculated by summing all the scores and then dividing by the number of scores. Now this formula can be written two different ways depending on whether we’re talking about the population or a sample, but the calculation is exactly the same. When we talk about populations we often use Greek letters and we refer to these as parameters and when we talk about samples we generally don’t use Greek letters and we refer to these as statistics. Now usually we don’t have access to actually studying our entire population and that means we’re going to use these statistics from samples to try to estimate the parameters of the population. So the population mean is referred to with the Greek letter mu and the formula would be mu equals Sigma x divided by uppercase n and for a sample mean we refer to it as uppercase M or as X-bar and the formula would be X bar equals Sigma x divided by lowercase n. And the upper or lowercase n here refers to whether we’re talking about the number of scores in the whole population uppercase or the number of scores in our sample, lowercase n.
The Greek uppercase Sigma here refers to sum so Sigma X means we sum up all the scores for variable X and I recommend that you get used to these Greek symbols. They’re going to appear often and you want to master them while we’re working with fairly basic concepts so later when we get to advanced concepts they won’t represent an additional burden on your understanding.
As a little side tip here if you want to type these Greek symbols in a word processor instead of going to insert special character and then searching around for them, it’s easier to just change your font to symbol then you just type the first letter of the Greek name for the symbol using lower or uppercase, depending on the symbol. So to type mu I set my font to symbol and then just type a lowercase m or for Sigma I just type an uppercase S.
So let’s quickly practice taking a mean with a small sample of scores. Let’s say we had six students who just took a quiz out of 10, and the scores were 1, 4, 5, 6, 7, and 10. So we’d sum all these values for x and get 33 then we divide by the number of scores which is 6. And so we’d get X bar equals 5.5 One of the strengths of the mean is that it’s very sensitive and this is because all scores contribute to it. So if we change any one score or we just add or remove one score, this has the potential to influence the mean. In this example if we just change the 7 here to a 10 we see the mean also changes; now – bar equals 6.
This sensitivity of the mean is generally a good thing but it comes at a cost. The risk is if we have extreme scores or outliers. Now if we have a symmetrical distribution with extreme scores on the low and high end they tend to cancel each other out so it doesn’t really matter. But if we have an asymmetrical distribution with more high or more low scores this is going to pull the mean in one direction. Let’s imagine our six students took another quiz and the scores were 0, 9, 9, 9, 9, 9, and 9. So in this case when we calculate the mean we get 7.5
This is a bit misleading because five students scored above 7.5 and only one student scored below it. So those five students might look at the mean and say “I did better than average because I scored above 7.5” when in fact they each only outscored one student.
A practical example of this would be looking at something like income. A very small number of people make very high incomes, hundreds of millions of dollars per year, and what this does is it’s going to pull the mean income to be misleadingly high. So in order to avoid this misleading mean we might want to look at a different measure of central tendency such as the median.
Median comes from the Latin for middle and it represents the middle score in our data and this can make it more appropriate if we have an asymmetrical distribution. So to find the median we just line up all the scores in order and then we find which score is in the middle. If we have an odd number of scores then the median is going to be the score that’s right in the middle. So if we had five scores, let’s say 1, 3, 5, 7, and 9 then our median would be 5. There’s two scores below that and there’s two scores above that.
If you have a larger number of odd scores and you don’t want to count them all you can find the median by taking the number of scores and adding 1 and then dividing by 2. This tells you the position of your median. So if I had 27 responses I would say 27 plus 1 divided by 2 equals 14 and that means the 14th score will be my median.
Now if we have an even number of scores then there’ll be two scores that fall in the middle and so what we’ll do is we’ll take the mean of those two values and that will be our median. So if we had six scores like 2, 4, 5, 6, 8, 9 we would see that the two in the middle are five and six and so we would take the mean of those and get a median of 5.5. To find these two middle scores without counting everything the first one will be at the position of n divided by 2 and the second one will be at n plus 2 divided by 2. So if we had 50 scores then we would look at the median being the mean of the 25th score and the 26th score.
The median is just based on the position of the scores when they’re in order, it doesn’t take into account exactly how small or large each individual score is. Now this means it’s not sensitive to outliers. We could change our highest score to 9,000 and the median would still be 5.5. So the median avoids this sensitivity of the mean but it does it by ignoring most of the values of your scores. It’s only looking at the value of the middle score or the middle two scores and it’s ignoring all the other values other than saying well they’re below that point or above that point. And that means that the median is going to be much more limited in the kinds of analyses you can use it for compared to the mean.
If we return to our income example we can see that a few extremely high earners won’t change the median, so it might be a better representation of typical income. But it also limits us if we only know the median. If we knew the mean we could actually figure out the total wealth in our sample; we would take the mean and multiply it by the number of our sample and that would give us the total but with the median we couldn’t do that because we don’t know the values of the lower or higher scores. I’ll add that there is a more precise way of calculating the median other than just taking the mean of the middle two scores but it’s more complicated and it’s generally unnecessary. In almost all cases the normal method of just taking the mean of those two middle scores will be sufficient. But if you’re curious I’ll make a separate video explaining this more precise method.
Our last measure of central tendency is the mode. This is the most frequent or the most commonly occurring score in our data. So if we had quiz scores of 3, 5, 5, 7, 7, 9 then our mode would be 7 because it occurred three times, which is more frequent than any of the other scores. Of course not all our distributions will be unimodal, meaning they have one most frequently occurring score. They could be bimodal where they have two high frequency scores that are equally occurring or multimodal where there are three or more equally high frequency scores. It’s also possible to have a uniform distribution where every score occurs equally frequently and in that case there’s no mode. For nominal data the mode is the only central tendency option that we have because we can’t do calculations on things like color names or occupations.
But the mode really doesn’t tell us very much. It tells us what the most frequently occurring score is but it doesn’t tell us how much more frequent that score was. Was it the most frequent by a few scores or by hundreds of scores? We don’t really know if all we know is the mode.
A strength of the mode is that it represents actual scores of participants. The mean and the median are computed values and that means they can give us values that none of the participants actually had. So in our examples we had a mean of 7.5 or a median of 5.5 and yet none of the participants actually had those scores. Whereas with the mode the most frequent score will always be actual scores participants had and this might give us a better representation of what’s typical.
A clear example of this would be saying that the mean number of children in a family is 2.4 Of course, nobody actually has 2.4 children so it might make more sense to talk about the mode, say most families have two children. This might give us a better understanding of what the typical family actually looks like, rather than a computed value.
So which of these measures of central tendency is best? Hopefully you see that we can’t really answer that. It depends on what you’re collecting data on and it depends on the distribution of your responses. Hopefully you also see that just knowing a measure of central tendency doesn’t tell you very much. This is why we’re going to need some other descriptive statistics to think about our data.
I hope you found this helpful, if so, please like the video and subscribe and don’t forget to check out the hundreds of other psychology tutorials that I have on the channel. Thanks for watching!
