A normal distribution has a smooth, symmetric, bell-shaped curve. Its mean, median and mode coincide at mu. Total area is 1, and probability is area over an interval; a single exact value has probability zero. Write X N(mu,sigma2), where sigma>0. The mean fixes the centre. A larger standard deviation spreads the same total area more widely, so the peak is lower. For any normal distribution, approximately 68%, 95% and 99.7% of the area lies within one, two and three standard deviations of the mean. These are estimates; use technology when an accurate probability is required. For mu=50 and sigma=8, about 95% of values lie between 34 and 66. A probability sketch should label the mean, relevant bounds and shaded event; distinguish the height of the curve from an area probability.

A normal distribution has a smooth, symmetric, bell-shaped curve. Its mean, median and mode coincide at \(\mu\). Total area is \(1\), and probability is area over an interval; a single exact value has probability zero.

Write \(X\sim N(\mu,\sigma^2)\), where \(\sigma>0\). The mean fixes the centre. A larger standard deviation spreads the same total area more widely, so the peak is lower.

For any normal distribution, approximately \(68\%\), \(95\%\) and \(99.7\%\) of the area lies within one, two and three standard deviations of the mean. These are estimates; use technology when an accurate probability is required.

For \(\mu=50\) and \(\sigma=8\), about \(95\%\) of values lie between \(34\) and \(66\). A probability sketch should label the mean, relevant bounds and shaded event; distinguish the height of the curve from an area probability.

Normal Shape and Spread

Normal Shape and Spread