The shape of data and the normal curve
Read the shape of a histogram (symmetric or skewed), see how skew pulls the mean away from the median, and use the 68–95–99.7 rule for bell-shaped data.
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Guide
The shape of a distribution tells a story. If the left and right halves of the histogram mirror each other, it is symmetric, and the mean and the median are close. If one side stretches out in a long tail, the data are skewed, and the skew is named after the side of the tail, not the side of the hump.
The tail pulls the mean towards it, because the mean uses every value, while the median only uses the middle one. With a long right tail the mean is greater than the median; with a long left tail it is smaller.
Many measurements (heights, errors) have a symmetric bell shape, the normal distribution. If the mean is x̄ and the standard deviation is S, about 68% of the data lie between x̄ − S and x̄ + S, about 95% between x̄ − 2S and x̄ + 2S, and almost all (99.7%) between x̄ − 3S and x̄ + 3S.