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I want to talk for a moment about proportions. Proportions are percentages, but written as a decimal, like 0.5 for 50%. People often report these and think of them like averages. And I just want to make a point that they’re actually a little bit different. They’re not averages. And I thought I would use some data from an earlier image for us to see that. By the way, if you remember from our earlier module, I really don’t like pie charts. I have a pie chart on the screen. But I’m going to always advocate that you use bar graphs. So imagine we have this following data as a bar graph. Woo! Glad we got rid of that pie chart.
55.8
On the left here, we see a situation where 80% of this sample identifies as male and 20% identifies as female. So here we have, clearly, the average or the mode, because it’s a categorical variable. We don’t have numbers here. These are just categories. The mode is male. The average score is male. And it’s 80%. On the right, we have a different data set. Now we have just over 50% of the sample as male, and the mode is also male. So here we have two situations where the average or typical score is male. That’s our summary statistic, our average for categorical variable. But the proportions are vastly different. Right?
101.5
And in the left case, clearly it’s the majority by a large margin. On the right, they’re almost equal. Proportions are useful for understanding where different areas of your data are. For example, what proportion of the sample is male or female, et cetera. However, I just want to point out that they are not averages. They are actually their own thing. If you get into more advanced statistical analyses, there are different kinds of statistics that we would have to use for proportions. If you’re wanting to do different kinds of statistical tests, with proportions as opposed to averages, there’s different equations. Everything changes when we’re dealing with proportions. We’re not going to learn all those tools in this class.
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But I want you to make that really clear in your notes that, in fact, when we’re dealing with proportions, it is a different kind of thing than an average. It’s useful, and in fact, I would always recommend calculating them, especially with categorical data. But they are not the same as averages. [LOGO MUSIC PLAYING]
Now that we understand the basics of Sigma notation, let’s begin with some statistical formulas. The first one we’re going to look at is how to calculate the average of a set of data.

In statistics, we call the average the mean and it has this symbol: (bar{x}). When we put a bar over any variable, we are calculating the average of that data set.

Simply put, to calculate the average of a set of data, we add up (sum or (Sigma)) each of the data points and then we divide it by the total number of data points that we have. The number of data points is represented by the letter (n).

Here is the formula we use to calculate the mean:
(bar{x} = frac{Sigma (x)}{n})

If we applied this formula to our age scores (of which we have 4 data points), it would look like this:
(bar{x} = frac{Sigma (x)}{n})
(therefore bar{x} = frac{30 + 21 + 59 + 45}{4})
(therefore bar{x} = frac{155}{4})
(therefore bar{x} = 38.75)

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Essential Mathematics for Data Analysis in Microsoft Excel

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