What Does Variance Measure?
Nathan Kamgang
We talked about averages in this article (link the article that talks about averages)
I assumed that you’ve read my article on average. From our explorations of mean or averages, we learnt that if the mean consider all datas from a sample of data. Imagine a machine that computes the mean score of an exam and you have to feed student scores one after the other. For each score you feed, the machine compute the mean on its screen. When you feed the data of student 1 who score 7/10 on our exam. The mean is computed by (7/1) (here use latex and show the formula for mean.) We take student 2 result of 3/10 and feed into the machine? what do you expect to happen? Our machine will now show(7+3/2) as mean. The mean score decreased. Now what if we add a 10/10. The mean score increased. So the mean tells emcopansess everyone performance. If you compare yourself to the mean, you know how well you’re considering eveyrone performance.
Comparing yourself to the mean tells you how far you are from the average student in the class. It’s an estimate that takes into account everyone’s results. More good students increase the mean. Poor performances decrease it.
Let’s say after feeding everyone one result the mean machine shows (x) as reading. As a student from my class who got a 7. you want to compare yourself to the mean. You’re +x above the mean. Likewise a student who failed with a score of y is -z fromt the mean. Note that the difference of a student from the mean is signed. Meaning it can be positive or negative.
In a class, each student has a score. But each student also has a distance from the average which is individualized as their score.
Each student from our class has a test score but also a difference from the mean. As a teacher you may like to know the aaverage difference from your mean. It will tell how much spread out your student result are. The higher that mean difference from the mean . The more spread out each individual student are. Remember that we said that the average tells encompasses everyone performance. So the average of difference from the mean encompasses everone deviation, difference or spread from that mean(Diagram showign difference from that mean) taking as sticks and averaged.
So if the class average is 6.7 and you score an 8, then you are 1.3 points above the average. Likewise, if you scored a 3, then you’re 3.7 points below average.
We are interested in how far everyone is from this mean, right? But what does “how far” mean if some people are below and above? We have 2 principal ways to solve this problem:
We consider the absolute value of those distances. Meaning if you’re 1.3 above, we just say that your distance is 1.3, and if you’re 3.7 below, your distance is just 3.7. We care about absolute distances. Another way is to square that difference. You know that a number times itself is always positive, right? So if your distance is -1.3, then your squared distance is -1.3 times -1.3, which always gives a positive number.
Everyone knows how to multiply, but does everyone know how to manipulate absolute values? It’s easier to deal with the squared distance for this reason.
Now comes the variance. When you add all the squared distances for everyone and you average them, you get the variance.
Table Example:
| Student | Score | Distance from Mean (6.7) | Squared Distance |
|---|---|---|---|
| Alice | 8 | 1.3 | 1.69 |
| Bob | 3 | -3.7 | 13.69 |
| Carol | 7 | 0.3 | 0.09 |
| David | 5 | -1.7 | 2.89 |
| Eva | 10 | 3.3 | 10.89 |
Variance = (1.69 + 13.69 + 0.09 + 2.89 + 10.89) ÷ 5 = 5.85
Wait, what? This variance was supposed to measure distance from the mean, but why is it this big? Because we squared everything, remember. So we can kind of do the reverse of squaring a number, which is taking the square root or finding the number which, when squared, gives our variance. That is called the standard deviation. It has a direct interpretation with our mean.
Standard Deviation = √5.85 = 2.42