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Q: What does small standard deviation signify?

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No, if the standard deviation is small the data is less dispersed.

That there is quite a large amount of variation between the observations.

No. A small standard deviation with a large mean will yield points further from the mean than a large standard deviation of a small mean. Standard deviation is best thought of as spread or dispersion.

When the sample size is small

Standard Deviation tells you how spread out the set of scores are with respects to the mean. It measures the variability of the data. A small standard deviation implies that the data is close to the mean/average (+ or - a small range); the larger the standard deviation the more dispersed the data is from the mean.

The standard deviation is the standard deviation! Its calculation requires no assumption.

The standard deviation of the population. the standard deviation of the population.

The standard deviation is 0.

Information is not sufficient to find mean deviation and standard deviation.

A large standard deviation means that the data were spread out. It is relative whether or not you consider a standard deviation to be "large" or not, but a larger standard deviation always means that the data is more spread out than a smaller one. For example, if the mean was 60, and the standard deviation was 1, then this is a small standard deviation. The data is not spread out and a score of 74 or 43 would be highly unlikely, almost impossible. However, if the mean was 60 and the standard deviation was 20, then this would be a large standard deviation. The data is spread out more and a score of 74 or 43 wouldn't be odd or unusual at all.

A small sample and a large standard deviation

The standard deviation in a standard normal distribution is 1.

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