Does Standard Deviation Increase With Sample Size

Our next goal is to determine how the size of the sample affects the variability nosotros meet in sample means.

Example

Sample Size Affects Variability of Sample Ways

We assumed that the population of individual babies has a hateful of µ = 3,500 grams and a standard deviation of σ = 500 grams. We selected a random sample of babies to test our assumptions near the population.

We saw previously that for this population of babies, it was not surprising to see a random sample of ix babies with a mean nascency weight of 3,400 grams. So a sample with a mean of iii,400 does non advise that the town'south mean nativity weight is less than 3,500 grams this yr.

What if we increase the sample size? Will our conclusion change? That is, if the hateful nascency weight of 3,400 grams comes from a larger sample of babies, does the sample provide stronger prove that the boondocks'southward mean nascency weight is less than 3,500 grams?

To investigate this question, nosotros ran the simulation for different sample sizes. For each sample size, nosotros collected i,000 random samples and recorded the sample means.

Three histograms with sample sizes of 9, 25, and 100. All have a center of around 3,500. The spread of each histogram gets smaller as sample size increases.

When nosotros compare the histograms of sample means, we detect the following:

Eye: The eye is not affected past sample size. The mean of the sample means is always approximately the aforementioned as the population mean µ = three,500.
Spread: The spread is smaller for larger samples, and so the standard divergence of the sample means decreases as sample size increases. This is not surprising considering we observed a like tendency with sample proportions.
Shape: The sampling distributions all appear approximately normal. This is not surprising because the distribution of birth weights in the population has a normal shape.

Based on the histograms, information technology appears that sample size will change our decision about the population's mean birth weight this twelvemonth. Suppose our sample mean of 3,400 grams came from a random sample of 100 babies. Ways from samples this large did not vary much. We marked this sample result in a histogram for samples of size 100.

Sample mean of 3,400 marked on a histogram of sample means taken from a sample size of 100. Only 50 out of 1,000 of the random samples have a sample mean of 3,400.

For n = 100, a sample mean of 3,400 grams is an unlikely effect. Information technology gives adequately strong testify that the population'southward hateful birth weight is less than 3,500 grams.

From avant-garde probability theory, nosotros have a probability model for the sampling distribution of sample ways. The model reinforces what nosotros have already observed about the heart and gives more precise data about the relationship between sample size and spread.

Suppose a population has a mean µ and a standard deviation of σ. The distribution of all possible sample means from this population will accept a hateful of µ and a standard divergence of [latex]σ\text{}/\sqrt{north}[/latex].

Permit's compare and contrast what we know near the sampling distributions for sample means and sample proportions.

Nosotros investigate the conditions that guarantee a normal sampling distribution for sample ways on the next page.