{"id":677,"date":"2020-07-27T14:13:05","date_gmt":"2020-07-27T18:13:05","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=677"},"modified":"2025-05-07T17:55:40","modified_gmt":"2025-05-07T21:55:40","slug":"6-2-distribution-of-the-sample-mean","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/6-2-distribution-of-the-sample-mean\/","title":{"raw":"6.2 Distribution of the Sample Mean","rendered":"6.2 Distribution of the Sample Mean"},"content":{"raw":"Suppose the variable of interest is X and the population consists of N individuals. The possible values of X are the different measurements for each individual in the population. For example, suppose the variable of interest is X=height and the population is the N = 60 students in our class. The number N = 60 is called the population size<\/strong>. Suppose we measure each student's height and draw a histogram of those N = 60 measurements. In that case, the resulting distribution is the population distribution<\/strong>, that is, the distribution of the random variable X. The average height of all 60 students is the population mean [latex]\\mu[\/latex].\r\n\r\nWe often use the sample mean [latex]\\bar{X}[\/latex] to estimate the population mean [latex]\\mu[\/latex]. However, since the observed value of [latex]\\bar{X}[\/latex] varies from sample to sample, it is helpful to know the typical accuracy of this estimator. For example, how confident are we that the error in estimating [latex]\\mu[\/latex] by [latex]\\bar{x}[\/latex] is at most 2 cm? To answer this kind of question, we need to know the distribution of the sample mean [latex]\\bar{X}[\/latex].\r\n\r\nFor a population of size N<\/em>, if we take a sample of size n<\/em>, there are [latex]\\binom{N}{n}[\/latex] distinct samples, each of which gives one possible value of the sample mean [latex]\\bar x[\/latex]. The [latex]\\binom{N}{n}[\/latex] values of [latex]\\bar{x}[\/latex] give the distribution of the sample mean [latex]\\bar{X}[\/latex], which is also called the sampling distribution of the sample mean. A histogram of the [latex]\\binom{N}{n}[\/latex] values of [latex]\\bar{x}[\/latex] shows the distribution of [latex]\\bar{X}[\/latex].\u00a0 However, [latex]\\binom{N}{n}[\/latex] is often so large that we are unable to consider all possible samples of size n directly. Fortunately, we can still obtain a reasonable approximation of the distribution of [latex]\\bar{X}[\/latex] by obtaining a large number of random samples, say 10,000, computing each sample mean, and drawing a histogram based on our sample of the sample means. For example, if the population size is N = 60 and the sample size is n = 5, there are [latex]\\binom{N}{n} = _{60}C_5 = 5,461,512[\/latex] different samples, many of which have different values of [latex]\\bar{x}[\/latex]. Drawing a histogram of these 5,461,512 [latex]\\bar{x}[\/latex] values gives the distribution of the sample mean [latex]\\bar{X}[\/latex], with sample size n = 5. Moreover, the sampling distribution of the sample mean [latex]\\bar{X}[\/latex] can be described in three aspects: centre, spread (variation), and shape.\r\n

6.2.1 Mean and Standard Deviation of the Sample Mean<\/h2>\r\nLet's consider a population consisting of 5 students. Suppose their heights (in cm) are [latex]x_1 = 155, x_2= 165, x_3=175, x_4=185, x_5=195[\/latex]. The population size is N<\/em>=5 and the population mean [latex]\\mu[\/latex] and population standard deviation [latex]\\sigma[\/latex] are: [latex]\\begin{align*} \\mu &= \\frac{\\sum x_i}{N} \\\\ \u00a0&= \\frac{155+165 + 175 + 185 +195}{5} \\\\ &= 175, \\\\ \\sigma &= \\sqrt{ \\frac{ \\sum (x_i - \\mu )^2 }{N} } \\\\ &= \\sqrt{\\frac{(155-175)^2 + (165 -175)^2 + (175 - 175)^2 + (185 - 175)^2 + (195-175,)^2} {5} } \\\\ &= 14.14. \\end{align*}[\/latex]\r\n\r\nConsider a simple random sample of size n<\/em> = 2, which means randomly picking two students from this population of five students. n<\/em> = 2 \u00a0is called the sample size<\/strong>. The number of ways we can pick two students out of five is [latex]_5C_2 = \\binom{5}{2} = 10[\/latex]. For example, one possible sample is [latex]\\{x_1, x_2\\}[\/latex] which gives a value of the sample mean,\r\n

[latex]\\bar{x} = \\frac{x_1 + x_2}{2} = \\frac{155+ 165}{2} = \u00a0160[\/latex].<\/p>\r\nAnother possible sample is [latex]\\{x_1, x_3 \\}[\/latex]\u00a0 and the corresponding value of the sample mean is:\r\n

[latex] \\bar{x} = \\frac{x_1 + x_3}{2} = \\frac{155+ 175}{2} = 165. [\/latex]<\/p>\r\nTable 6.1 lists all possible samples of sample size n<\/em> = 2, 3, 4 and their corresponding sample mean values. The mean and standard deviation of the sample mean of all possible sample sizes are also given in the table.<\/a>\r\n\r\n[caption id=\"attachment_721\" align=\"aligncenter\" width=\"851\"]\"A Table 6.1<\/strong>: Sample Means of All Possible Samples of Sample Size n=2, 3, 4. [Image Description (See Appendix D Table 6.1)<\/a>][\/caption]The mean and standard deviation of the sample mean [latex]\\bar{X}[\/latex] are denoted as [latex]\\mu_{\\bar{X}}[\/latex] and [latex]\\sigma_{\\bar{X}}[\/latex] respectively. When the sample size [latex]n=2[\/latex], Table 6.1 shows 10 possible values of the sample mean: [latex]160, 165, \\cdots, 185, 190[\/latex]; there is one value of 160 and two values of 180, giving the probabilities of [latex]\\frac{1}{10}[\/latex] and [latex]\\frac{2}{10}[\/latex] observing these two values respectively. The probability distribution and distribution histogram of the sample mean [latex]\\bar{X}[\/latex] with [latex]n=2[\/latex] are:<\/a>\r\n\r\n[caption id=\"attachment_720\" align=\"aligncenter\" width=\"797\"]\"A Figure 6.1<\/strong>: Probability Distribution and Probability Histogram of Sample Mean for n=2. [Image Description (See Appendix D Figure 6.1)<\/a>][\/caption]The mean and the standard deviation of the sample mean with n<\/em> = 2\u00a0are:\r\n

[latex]\\begin{align*} \\mu_{\\bar{X}} &= \\frac{160 + 165 + 170 + 175+ 170 + 175 + 180 + 180 + 185 + 190}{10} \\\\ &= 175, \\\\ \\sigma_{\\bar{X}} &= \\sqrt{ \\frac{\\sum (\\bar{x} - \\mu_{\\bar{X}})^2}{N}} \\\\ &= \\sqrt{ \\frac{ (160-175)^2 + (165-175)^2 + ... + (185 - 175)^2 + (190 - 175)^2}{10} } \\\\ &= 8.66. \\end{align*}[\/latex]<\/p>\r\nWhen the sample size is n<\/em> = 3, the mean and the standard deviation of the sample mean are:\r\n

[latex]\\begin{align*} \\mu_{\\bar{X}} &= \\frac{165 + 168.33 + 171.67 + 171.67+ 175 + 178.33 + 175 + 178.33 + 181.67 + 185}{10} \\\\ &= 175, \\\\ \\sigma_{\\bar{X}} &= \\sqrt{ \\frac{\\sum (\\bar{x} - \\mu_{\\bar{X}})^2}{N}} \\\\ &= \\sqrt{ \\frac{ (160-175)^2 + (168.33-175)^2 + ... + (185 - 175)^2 + (190 - 175)^2}{10} } \\\\ &= 5.77. \\end{align*}[\/latex]<\/p>\r\nWhen the sample size is n<\/em> = 4, the mean and the standard deviation of the sample mean are:\r\n

[latex]\\begin{align*} \\mu_{\\bar{X}} &= \\frac{170 +172.5 +175 +177.5 + 180 }{5} \\\\ &= 175, \\\\ \\sigma_{\\bar{X}} &= \\sqrt{ \\frac{\\sum (\\bar{x} - \\mu_{\\bar{X}})^2}{N}} \\\\ &= \\sqrt{ \\frac{ (170-175)^2 + (172.5-175)^2 + (175 - 175)^2 + (177.5 - 175)^2 + (180 - 175)^2}{5} } \\\\ &= 3.54. \\end{align*}[\/latex]<\/p>\r\nThe above results show that the mean of the sample mean equals the population mean regardless of the sample size, i.e., [latex]\\mu_{\\bar{X}} = \\mu[\/latex], while the standard deviation of the sample mean decreases when the sample size n<\/em> increases. It can be shown that when sampling without replacement from a finite population, like those listed in Table 6.1,\r\n

[latex] \\sigma_{\\bar{X}} = \\sqrt{ \\frac{N-n}{N-1} } \\times \\frac{\\sigma}{\\sqrt{n}}. [\/latex]<\/p>\r\nIf we instead sample with replacement from a finite population, the standard deviation of the sample mean is\r\n

[latex] \\sigma_{\\bar{X}} = \\frac{\\sigma}{\\sqrt{n}}. [\/latex]<\/p>\r\nNote<\/strong>: If we sample without replacement, [latex] \\sigma_{\\bar{X}}[\/latex] is approximately equal to [latex]\\frac{\\sigma}{\\sqrt{n}}[\/latex], as long as the sample size n<\/em> is much smaller than the population size N<\/em>. For simplicity of notation, we only focus on the sample without replacement case for the distribution of the sample mean in the remaining chapters.\r\n

\r\n

Key Facts: Mean and Standard Deviation of the Sample Mean [latex]\\color{white}{\\bar{X}}[\/latex]<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nFor samples of size n<\/em>,\r\n