What a high or low standard deviation actually tells you
Quick answer: Standard deviation measures how spread out values are around the mean, in the same units as the data. A small standard deviation means values cluster close to the average; a large one means they are widely scattered. It can never be negative.
The standard deviation summarizes the typical distance between a value and the mean. Because it is expressed in the original units — dollars, centimetres, points — you can read it directly: a standard deviation of 5 points means values sit, on average, about 5 points away from the mean.
There is no universally "good" value; it depends entirely on context. Two classes can share an average test score of 75, but if one has a standard deviation of 3 and the other 15, the first class is far more consistent while the second has both strong and struggling students. A low standard deviation signals consistency or precision; a high one signals variability. What counts as high or low only makes sense relative to the mean and the situation — which is what the coefficient of variation captures.
For data that is roughly bell-shaped (normal), the empirical rule gives standard deviation an intuitive meaning: about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. So if exam scores average 70 with a standard deviation of 8, roughly 95% of students scored between 54 and 86. The rule is only an approximation and applies best to symmetric, mound-shaped data.
Data: 5, 7, 3, 7, 8, 5, 9, 6, 4, 6 — mean = 6. The sample standard deviation is about 1.83, so a typical value sits roughly 1.8 units from the mean of 6. Because the spread is small relative to the mean, this data is fairly consistent.
It tells you how far values typically fall from the mean. Small means tightly clustered; large means widely spread.
No. It is the square root of an average of squared deviations, so it is always zero or positive. It equals zero only when all values are identical.
Not inherently. Lower variability is desirable for things like manufacturing tolerances, while higher variability may be expected or even wanted in other contexts. It depends on what you are measuring.