{"id":309,"date":"2020-06-29T15:33:01","date_gmt":"2020-06-29T19:33:01","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=309"},"modified":"2025-05-07T17:40:52","modified_gmt":"2025-05-07T21:40:52","slug":"2-5-descriptive-measures-for-population-and-sample","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/2-5-descriptive-measures-for-population-and-sample\/","title":{"raw":"2.5 Descriptive Measures for Population and Sample","rendered":"2.5 Descriptive Measures for Population and Sample"},"content":{"raw":"We summarize the descriptive measures for the population and for the sample in the following table. Note that a summation sign without indices means taking the sum of all observations in the data, e.g., the population mean [latex] \\mu = \\frac{\\sum_{i=1}^N x_i}{N} = \\frac{\\sum x_i}{N}[\/latex].\r\n\r\n\r\n\r\n
Population<\/strong>\r\n\r\nDefinition: The collection of all individuals under consideration in a study.\r\n\r\nPopulation size [latex]N[\/latex]= the total number of individuals in the population.\r\n\r\nPopulation mean [latex]\\mu[\/latex]: Suppose the measurement of each individual is [latex]x_1, x_2, \\cdots, x_N[\/latex], the population mean is defined as\r\n\r\n[latex]\\mu=\\frac{x_1+x_2+\\cdots+x_N}{N}=\\frac{\\sum_{i=1}^N x_i}{N}=\\frac{\\sum x_i}{N}.[\/latex]\r\n\r\nPopulation standard deviation, [latex]\\sigma[\/latex], is the square root of the population variance [latex]\\sigma^2[\/latex]. It is defined as\r\n\r\n[latex]\\sigma=\\sqrt{\\frac{\\sum_{i=1}^N (x_i-\\mu)^2}{N}}=\\sqrt{\\frac{\\sum (x_i-\\mu)^2}{N}}.[\/latex]\r\n\r\nThe following formula is helpful in calculating the population standard deviation,\r\n\r\n[latex]\\sigma=\\sqrt{\\frac{\\sum_{i=1}^N x^2_i}{N}-\\mu^2}=\\sqrt{\\frac{\\sum x^2_i}{N}-\\mu^2}[\/latex]\r\n\r\nA descriptive measure for a population, such as [latex]\\mu[\/latex] and [latex]\\sigma[\/latex], is called a parameter<\/em>.<\/td>\r\n\r\n

Sample<\/strong><\/p>\r\nDefinition: Part of or a subset of the population from which information is obtained.\r\n\r\nSample size [latex]n[\/latex]= the total number of individuals in the sample.\r\n\r\nSample mean [latex]\\bar x[\/latex]: Suppose the measurements of the sample are [latex]x_1, x_2, \\cdots, x_n[\/latex], the sample mean is defined as\r\n\r\n[latex]\\bar x=\\frac{x_1+x_2+\\cdots+x_n}{n}=\\frac{\\sum_{i=1}^n x_i}{n}=\\frac{\\sum x_i}{n}.[\/latex]\r\n\r\nSample standard deviation, [latex]s[\/latex], is the square root of the sample variance [latex]s^2[\/latex]. It is defined as\r\n\r\n[latex]s=\\sqrt{\\frac{\\sum_{i=1}^n (x_i-\\bar x)^2}{n-1}}=\\sqrt{\\frac{\\sum (x_i-\\bar x)^2}{n-1}}.[\/latex]\r\n\r\nThe following formula is helpful in calculating the sample standard deviation,\r\n\r\n[latex]s=\\sqrt{\\frac{\\sum_{i=1}^n x^2_i-\\frac{(\\sum x_i)^2}{n}}{n-1}}=\\sqrt{\\frac{\\sum x^2_i-\\frac{(\\sum x_i)^2}{n}}{n-1}}[\/latex]\r\n\r\nA descriptive measure for a sample, such as [latex]\\bar x[\/latex] and [latex]s[\/latex], is called a statistic<\/em>.<\/td>\r\n<\/tr>\r\n<\/thead>\r\n<\/table>\r\n ","rendered":"

We summarize the descriptive measures for the population and for the sample in the following table. Note that a summation sign without indices means taking the sum of all observations in the data, e.g., the population mean [latex]\\mu = \\frac{\\sum_{i=1}^N x_i}{N} = \\frac{\\sum x_i}{N}[\/latex].<\/p>\n\n\n\n
Population<\/strong><\/p>\n

Definition: The collection of all individuals under consideration in a study.<\/p>\n

Population size [latex]N[\/latex]= the total number of individuals in the population.<\/p>\n

Population mean [latex]\\mu[\/latex]: Suppose the measurement of each individual is [latex]x_1, x_2, \\cdots, x_N[\/latex], the population mean is defined as<\/p>\n

[latex]\\mu=\\frac{x_1+x_2+\\cdots+x_N}{N}=\\frac{\\sum_{i=1}^N x_i}{N}=\\frac{\\sum x_i}{N}.[\/latex]<\/p>\n

Population standard deviation, [latex]\\sigma[\/latex], is the square root of the population variance [latex]\\sigma^2[\/latex]. It is defined as<\/p>\n

[latex]\\sigma=\\sqrt{\\frac{\\sum_{i=1}^N (x_i-\\mu)^2}{N}}=\\sqrt{\\frac{\\sum (x_i-\\mu)^2}{N}}.[\/latex]<\/p>\n

The following formula is helpful in calculating the population standard deviation,<\/p>\n

[latex]\\sigma=\\sqrt{\\frac{\\sum_{i=1}^N x^2_i}{N}-\\mu^2}=\\sqrt{\\frac{\\sum x^2_i}{N}-\\mu^2}[\/latex]<\/p>\n

A descriptive measure for a population, such as [latex]\\mu[\/latex] and [latex]\\sigma[\/latex], is called a parameter<\/em>.<\/td>\n

\n

Sample<\/strong><\/p>\n

Definition: Part of or a subset of the population from which information is obtained.<\/p>\n

Sample size [latex]n[\/latex]= the total number of individuals in the sample.<\/p>\n

Sample mean [latex]\\bar x[\/latex]: Suppose the measurements of the sample are [latex]x_1, x_2, \\cdots, x_n[\/latex], the sample mean is defined as<\/p>\n

[latex]\\bar x=\\frac{x_1+x_2+\\cdots+x_n}{n}=\\frac{\\sum_{i=1}^n x_i}{n}=\\frac{\\sum x_i}{n}.[\/latex]<\/p>\n

Sample standard deviation, [latex]s[\/latex], is the square root of the sample variance [latex]s^2[\/latex]. It is defined as<\/p>\n

[latex]s=\\sqrt{\\frac{\\sum_{i=1}^n (x_i-\\bar x)^2}{n-1}}=\\sqrt{\\frac{\\sum (x_i-\\bar x)^2}{n-1}}.[\/latex]<\/p>\n

The following formula is helpful in calculating the sample standard deviation,<\/p>\n

[latex]s=\\sqrt{\\frac{\\sum_{i=1}^n x^2_i-\\frac{(\\sum x_i)^2}{n}}{n-1}}=\\sqrt{\\frac{\\sum x^2_i-\\frac{(\\sum x_i)^2}{n}}{n-1}}[\/latex]<\/p>\n

A descriptive measure for a sample, such as [latex]\\bar x[\/latex] and [latex]s[\/latex], is called a statistic<\/em>.<\/td>\n<\/tr>\n<\/thead>\n<\/table>\n

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