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Measures of Location and Dispersion Central Tendency

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Mean, Median, Mode Of Statistical Data

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where µ is the population mean. X is a particular value. indicates the operation of adding. N is the number of observations in the population. Population Mean The population mean is the sum of the population values divided by the number of population values:

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PRACTICEPRACTICE Parameter: a measurable characteristic of a population. The scores of 4 low-achieving students is compared: 56; 23; 42; and 73. Find the mean score. The mean is: ( )/4 = 48.5 Parameter: a measurable characteristic of a population. The scores of 4 low-achieving students is compared: 56; 23; 42; and 73. Find the mean score. The mean is: ( )/4 = 48.5

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Sample Mean The sample mean is the sum of the sample values divided by the number of sample values: where X is for the sample mean n is the number of observations in the sample

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PRACTICEPRACTICE A sample of five part-time teacher salaries is compared: $15,000, $15,000, $17,000, $16,000, and $14,000. Find the mean bonus for these five teachers. Since these values represent a sample size of 5, the sample mean is (14, , , , ,000)/5 = $15,400. A sample of five part-time teacher salaries is compared: $15,000, $15,000, $17,000, $16,000, and $14,000. Find the mean bonus for these five teachers. Since these values represent a sample size of 5, the sample mean is (14, , , , ,000)/5 = $15,400.

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Properties of the Arithmetic Mean 1.Every set of interval and ratio-level data has a mean 2.All the values are included in computing the mean 3.A data set will only have one unique mean 4. The mean is a useful measure for comparing two or more populations (we'll discuss this one later when we get to t- tests and analysis of variance) and… 1.Every set of interval and ratio-level data has a mean 2.All the values are included in computing the mean 3.A data set will only have one unique mean 4. The mean is a useful measure for comparing two or more populations (we'll discuss this one later when we get to t- tests and analysis of variance) and…

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Properties of the Arithmetic Mean 5.The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero.

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PRACTICEPRACTICE What is the mean for: 3, 8, and 4? The mean is 5. Illustrate the fifth property: What is the mean for: 3, 8, and 4? The mean is 5. Illustrate the fifth property: (3-5) + (8-5) + (4-5) = = 0.

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The Median Median: The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. There are as many values above the median as below it in the data array. Note: For an even set of numbers, the median will be the arithmetic mean of the two middle numbers. Median: The midpoint of the values after they have been ordered from the smallest to the largest, or the largest to the smallest. There are as many values above the median as below it in the data array. Note: For an even set of numbers, the median will be the arithmetic mean of the two middle numbers.

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PRACTICEPRACTICE Compute the median for the following data: The age of a sample of five college students is: 21, 25, 19, 20, and 22. Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The height of four basketball players, in inches, is 76, 73, 80, and 75. Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is Compute the median for the following data: The age of a sample of five college students is: 21, 25, 19, 20, and 22. Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. The height of four basketball players, in inches, is 76, 73, 80, and 75. Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5.

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Properties of the Median 1.There is a unique median for each data set. 2.It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. 3.It can be computed for ratio-level, interval-level, and ordinal-level data. 4.It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. 1.There is a unique median for each data set. 2.It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. 3.It can be computed for ratio-level, interval-level, and ordinal-level data. 4.It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

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The Mode The mode is the value of the observation that appears most frequently. EXAMPLE 6: the exam scores for ten students are: 83, 89, 84, 75, 99, 87, 83, 75, 83, 87. Find the mode. Since the score of 83 occurs the most, the modal score is 83. The mode is the value of the observation that appears most frequently. EXAMPLE 6: the exam scores for ten students are: 83, 89, 84, 75, 99, 87, 83, 75, 83, 87. Find the mode. Since the score of 83 occurs the most, the modal score is 83.

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Measures of Dispersion Calculating Data SPREAD

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Measures of Dispersion The range is the difference between the highest and lowest values in a set of data. RANGE = Highest Value - Lowest Value PRACTICE: A sample of five accounting graduates revealed the following starting salaries: $22,000, $28,000, $31,000, $23,000, $24,000. The range is $31,000 - $22,000 = $9,000. The range is the difference between the highest and lowest values in a set of data. RANGE = Highest Value - Lowest Value PRACTICE: A sample of five accounting graduates revealed the following starting salaries: $22,000, $28,000, $31,000, $23,000, $24,000. The range is $31,000 - $22,000 = $9,000.

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Population Variance The population variance is the arithmetic mean of the squared deviations from the population mean. The population variance is the arithmetic mean of the squared deviations from the population mean.

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PRACTICEPRACTICE The ages of the Jones family are 2, 18, 34, and 42 years. What is the population variance? 3-35

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Population Variance Alternative formulas for the population variance are:

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The Population Standard Deviation The population standard deviation (σ) is the square root of the population variance. For the PRACTICE, what is the standard deviation for the ages? (variance = 236) the population standard deviation is (square root of 236). The population standard deviation (σ) is the square root of the population variance. For the PRACTICE, what is the standard deviation for the ages? (variance = 236) the population standard deviation is (square root of 236).

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Sample Variance The sample variance estimates the population variance. Conceptual Formula Computational Formula

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PRACTICEPRACTICE ( /5)/5-1 s 2 = 21.2/(5-1) = 5.3 ( /5)/5-1 s 2 = 21.2/(5-1) = 5.3 A sample of five hourly wages for various jobs on campus is: $7, $5, $11, $8, $6. Find the variance.

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Sample Standard Deviation The sample standard deviation is the square root of the sample variance. Find the sample standard deviation for example 14. s 2 = 5.3 s = 2.30 The sample standard deviation is the square root of the sample variance. Find the sample standard deviation for example 14. s 2 = 5.3 s = 2.30

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Interpretation and Uses of the Standard Deviation Empirical Rule: For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within +/- one SD of the mean approximately 95% of the observations will lie within +/- 2 SD of the mean approximately 99.7% within +/- 3 SD of the mean.

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