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The Significance of Standard Deviation Consider the volumes of liquid in different cans of a particular brand of soft drink. The distribution of volumes is symmetrical and bell-shaped. This is due to natural variation produced by the machine which has been set to produce a particular volume. Random or chance factors cause roughly the same number of cans to be overfilled as underfilled. The resulting bell-shaped distribution is called the If a large sample from a typical bell-shaped data distribution is taken, what percentage of the data values would lie between 𝑥̅ − 𝑠 and 𝑥̅ + 𝑠? It can be shown that for any measured variable from any population that is normally distributed, no matter the values of the mean and standard deviation: Approximately of the population will measure between deviation either side of the mean Approximately of the population will measure between deviation either side of the mean Approximately of the population will measure between deviation either side of the mean standard standard standard Example A sample of 200 cans of peaches was taken from a warehouse and the contents of each can measured for net weight. The sample mean was 486g with standard deviation 6.2g. What proportion of the cans might lie within: a. 1 standard deviation from the mean b. 3 standard deviations from the mean The Normal Curve The smooth curve that models normally distributed data is asymptotic to the horizontal axis, so in theory there are no limits within which all the members of the population will fall. In practice, however, it is rare to find data outside of 3 standard deviations from the mean, and exceptionally rare to find data beyond 5 standard deviations from the mean. Note that the position of 1 standard deviation either side of the mean corresponds to the point where the normal curve changes from a concave to a convex curve. Chapter 23 – Descriptive Statistics IB Mathematics