| Sampling Distribution Properties |
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Sampling from a normal distribution |
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| According to theory (The Central Limit Theorem) means of samples from a normal distribution will be normally distributed. This statement applies to a large number of samples. One particular collection of samples will probably depart from normality, especially if the sample size is small or fewer samples are taken. In both the sampling distributions below, the number of samples is 50. They were taken from a normal population. |
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Figure 1. Means of 50 samples.
Sample size 5.
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Figure 2. Means of 50 samples.
Sample size 25.
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#30 a) Using figures 1 and 2 as evidence, how did changing sample size affect the variability of the sample means? |
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b) Why did the vertical scale change? |
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| Sampling from a skewed distribution |
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The distribution in figure 3 below represents 405 length-of-stay records for a hospital unit, in days. The histogram is clearly from a distribution which is not normal. It is skewed to the right. (It is also not normal because the categories are discrete.)
Notice that the data are individual observations. |
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| Next we will sample the length of stay data, and calculate the mean of each sample. Then we will plot the distribution of the samples means we obtained. The sampling distributions are shown |
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Figure 3. A skewed distribution.
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The sampling distributions from the length of stay data are more balanced than the parent data. With larger sample sizes the sampling distribution appears increasingly normal, and the variability decreases. |
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Figure 4. Means of 50 samples from the LOS data. Sample size 5.
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Figure 5. Means of 50 samples from the LOS data. Sample size 5.
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The basic normality of sampling distributions is a tremendous gift! It means all the properties of the normal distribution can be applied to sampling distributions, even if the parent distribution is not exactly normal! This is the core principle of statistical process control. Of course some cautions apply.
Sample means from a normal distribution are normally distributed. Sample means from an non-normal distribution will be close to normally distributed if the sample size is large and the parent distribution is balanced in shape. The means of samples of size 5 from the LOS data in figure 4 are certainly not normal. They are not even balanced. This topic is discussed in the next page on sample means. |
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