module 20 distribution of sample means 5 of 12 our next goal is to determine how the size of the sample affects the variability we see in Sample means example sample size affects variability of sample means we assume that the population of individual babies this year has the mean birth weight of mu equal to 3,500 G and a standard deviation of 500 gram we selected a random sample of babies to test our assumptions about the population we saw previously that for this population of babies it was not surprising to see a random sample of nine babies with a mean birth weight of 3,400 G so a sample with a mean of 3,400 does not suggest that the town's mean birth weight is less than 3,500 gram this year but what if we increase the sample size will our conclusion change that is if the mean birth weight of 3,400 G comes from a larger sample of babies does the sample provide stronger evidence that the Town's mean birth weight is less than 3,500 G to investigate this question we ran the simulation for different sample sizes Nal to 9 babies n equal to 25 babies and N equal to 100 babies in the sample for each sample size we collected 1,000 random samples of the sample size and recorded the sample means to generate the sampling distribution so here we have sample means for samples of nine babies um and you notice that the centers have about 3,500 but one standard deviation away is over here lower and over here higher um here's a sample mean still at 3,500 but now we have samples of 25 babies and notice that the standard deviation is a little bit one standard deviation away is a little bit closer to the mean than it was with a sample of nine and then if you get samples of 100 babies notice how we are even closer one standard deviation away is even closer to the mean of 3,500 100 so when we compare the histograms of sample means we notice the following for Center the center is not affected by sample size the mean of the sample means is always approximately the same as a population mean 3,500 spread the spread is smaller for larger samples so the standard deviation of a sample means decreases as sample size increases this is not surprising because we observed a similar Trend 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 history histograms it appears that the sample size will change our conclusion about the population's mean birth weight this year suppose our sample mean of 3,400 G came from a random sample of 100 babies means from samples this large did not vary much we marked the sample result in a histogram for samples of size 100 so in other words here's histogram for samples of 100 babies and this blue marker here this blue line is where the weight mean weight of 3,400 G is and notice that it's further away from the center the mean which is right in the middle here and it says for n equal to 100 a sample mean of 3,400 G is an unlikely result it gives fairly strong evidence that the population's mean birth weight is less than 3,500 G from Advanced probability Theory we have a probability model for the sampling distribution of sample means the model reinforces what we have already observed about the center and gives more precise information about the relationship between sample size and spread so the theoretical probability model for the sampling distribution of sample means suppose a population has a mean mu and standard deviation of Sigma the distribution of all possible sample means from this population we'll have a mean of mu and and a standard deviation of Sigma / the square root of the sample size n