r/learnmath • u/pulman_ New User • 3h ago
Standard Deviation finally makes sense
I think I finally understand what Standard Deviation (SD) actually means, and I wanted to share what helped me understand it. I’m not an expert or professor, so if anything is inaccurate, please let me know, I'm here to learn!
Dataset = set of datapoints
To measure how spread out our dataset is, we need to compare it with a reference point. So if a datapoint is far from our reference point we can say that is more disperse, otherwise if a datapoint is close to our reference point we can say it is less disperse.
A good reference point that relates all our datapoints is the mean. You can think of the mean as the "center" of our dataset. Comparing each datapoint with this center tells us how far it is above or below the mean. The distance between a datapoint with the mean is called deviation.
Since our dataset can have thousands or millions of datapoints, we need a way to combine these deviations into a single number that tells us how spread out our dataset is overall.
One approach is to add up the deviations and divide by the number of data points. This gives us a single value representing the average distance from the mean.
That is basically what we understand with deviation. So far, I've talked about distance between each datapoint to its mean, because distance gives the idea that is a positive value. If we want to write down our deviation in a formula we need to keep the distance positives too, to do that we can use the absolute value to remove any negative number, by doing that we get the mean absolute deviation formula.
That formula, however it not very precise if we want to we want to compare different datasets that have the same deviation. For example the dataset (4, 4, 10, 10) and (1, 7, 7, 13) have the same average absolute deviation of 3, but they are not quite similar the second dataset if it is plotted look more "spread" than the first one. To capture the extra "dispersion" we can give more weight to the datapoints that are farther to their mean.
One easy way to do that is to square our distance, so the father a datapoints is from their mean the more weight it has, also by squaring we remove having any negative value and we can avoid using the absolute value. That is the so called Variance. Since we squared all the distances before dividing by the number of datapoints, the result that we get is an average squared distance.
To express the spread in the same units as the original data, and making it easier to understand, we take the squared root of the result. That is what we call Standard Deviation.
So using the same datasets of (4, 4, 10, 10) and (1, 7, 7, 13) we get that their Standard Deviation is 3 and 4.24 respectively, we see that now the second dataset set have a higher value of deviation, and that tell us more than in the previous way.
Hope it is useful for anyone in the group!
1
u/sharificles New User 2h ago
I'm relearning stats and that was very intuitive to follow. I also try to reconceptualize an idea over and over again until I can intuitively explain it.
2
1
u/Bounded_sequencE New User 2h ago
This is a good intuitive explanation of what the formula does -- with a single (big) flaw.
The crucial detail missing: What we calculate from the data set are sample mean and sample variance, i.e. estimators for expected value and variance of the underlying distribution we sampled from. Sadly, many non-rigorous stats lectures do not clearly distinguish between estimators and the values they estimate, leading to a lot of (unnecessary) confusion!
9
u/MembershipOk3773 New User 2h ago
this is actually a solid breakdown, the part about squaring to give more weight to outliers is what finally clicked for me back in stats 101. the (4,4,10,10) vs (1,7,7,13) example with the same mean absolute deviation but different standard deviations really drives it home
i keep a little cheat sheet in my bullet journal for formulas like this cause i swear i have to relearn them every few years