what is the difference between what variance measures and what standard deviation measures?
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Variance is just the square of the standard deviation. They’re essentially a measure of the same thing.
They’re a measure of how spread out the data is. The smaller the standard deviation the more likely a point of data is close to the mean.
If you are familiar with a normal distribution (AKA Gaussian distribution or bell curve), the the standard deviation will tell you how peaked the curve is. A small standard deviation results in a narrow high peak, and a large standard deviation results wide low hill.
For a normal distribution (a very common distribution thanks in part to the central limit theorm) the standard deviation can be used to predict how much of a dataset is within any given range. ~68% of the dataset will be within 1 standard deviation of the mean, ~95% within 2 standard deviations, and ~99.7% within 3 standard deviations.
For example, in the USA the height of adult US males is approximately a normal distribution with a mean of 70 inches and a standard deviation of 3 inches. From this we can assume that 68% of adult US males will be between 67 and 73 inches tall, 95% will be between 64 and 76 inches tall, and 99.7% will be between 61 and 79 inches tall.
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