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Statistics

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The Mean and Median: Measures of Central Tendency

• The mean and the median are summary measures used to describe the most "typical" value in a set of values.

• Statisticians refer to the mean and median as measures of central tendency.

Median

• To find the median, we arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

Mean

• The mean of a sample or a population is computed by adding all of the observations and dividing by the number of observations.

• Population mean = μ = ΣX / N

• OR Sample mean = x = Σx / n

The Mean vs. the Median

• The median may be a better indicator of the most typical value if a set of scores has anoutlier. An outlier is an extreme value that differs greatly from other values.

• However, when the sample size is large and does not include outliers, the mean score usually provides a better measure of central tendency.

The Variance

• In a population, variance is the average squared deviation from the population mean, as defined by the following formula:

• σ2 = Σ ( Xi - μ )2 / N• where σ2 is the population variance, μ is the population

mean, Xi is the ith element from the population, and N is the number of elements in the population.

• Observations from a simple random sample can be used to estimate the variance of a population. For this purpose, sample variance is defined by slightly different formula, and uses a slightly different notation:

• s2 = Σ ( xi - x )2 / ( n - 1 )

The Standard Deviation

• The standard deviation is the square root of the variance. Thus, the standard deviation of a population is:

σ = sqrt [ σ2 ] = sqrt [ Σ ( Xi - μ )2 / N ]• Thus, the standard deviation of a Sample is: s = sqrt [ s2 ] = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ]

Problem 1A weatherman records the high temperature in Norfolk, Va. for each of the first 7 days of the year. The temperatures (in degrees fahrenheit) are 51, 49, 41, 53, 54, 68, 48. Over that time period, which of the following statements is true?I. The mean is 51. II. The mean is 52. III. The median is 53. IV. The median is 54.(A) I only (B) II only (C) III only (D) IV only (E) None is true

SolutionThe correct answer is (B). The mean score is computed from the equation:Mean score = Σx / n = (51 + 49 + 53 + 41 + 68 + 48 + 54) / 7 = 52Since there are an odd number of scores (7 temperatures), the median is the middle score. If we arrange the temperatures in order (from the coldest day to the hottest day), we get the following arrangement: 41, 48, 49, 51, 53, 54, 68. Thus, the middle score is 51, so the median is 51.

Problem 2Four friends take an IQ test. Their scores are 96, 100, 106, 114. Which of the following statements is true?I. The mean is 103. II. The mean is 104. III. The median is 100. IV. The median is 106.(A) I only (B) II only (C) III only (D) IV only (E) None is true

SolutionThe correct answer is (B). The mean score is computed from the equation:Mean score = Σx / n = (96 + 100 + 106 + 114) / 4 = 104Since there are an even number of scores (4 scores), the median is the average of the two middle scores. Thus, the median is (100 + 106) / 2 = 103.

Problem 3A sample consists of four observations: {1, 3, 5, 7}. What is the standard deviation?(A) 2.24 (B) 2.58 (C) 5.00 (D) 6.67 (E) None of the above

SolutionThe correct answer is (B). First, we need to compute the sample mean.x = ( 1 + 3 + 5 + 7 ) / 4 = 4Then we plug all of the known values into formula for the standard deviation of a sample, as shown below:s = sqrt [ Σ ( xi - x )2 / ( n - 1 ) ] s = sqrt { [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / ( 4 - 1 ) } s = sqrt { [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 3 } s = sqrt { [ 9 + 1 + 1 + 9 ] / 3 } = sqrt (20 / 3) = sqrt ( 6.67 ) = 2.58

Problem 4

A population consists of four observations: {1, 3, 5, 7}. What is the standard deviation?(A) 2.24 (B) 2.58 (C) 5.00 (D) 6.67 (E) None of the above

SolutionThe correct answer is (A). First, we need to compute the population mean.μ = ( 1 + 3 + 5 + 7 ) / 4 = 4Then, we plug all of the known values into formula for the standard deviation of a population, as shown below:σ = sqrt [ Σ ( Xi - μ )2 / N ] σ = sqrt { [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / 4 } σ = sqrt { [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 4 } σ = sqrt { [ 9 + 1 + 1 + 9 ] / 4 } = sqrt ( 20 / 4 ) = sqrt(5) = 2.24

Problem 5

A sample consists of four observations: {1, 3, 5, 7}. What is the variance?(A) 2.24 (B) 2.58 (C) 5.00 (D) 6.67 (E) None of the above

SolutionThe correct answer is (D). First, we need to compute the sample mean.x = ( 1 + 3 + 5 + 7 ) / 4 = 4Then we plug all of the known values into formula for the variance of a sample, as shown below:s2 = Σ ( xi - x )2 / ( n - 1 ) s2 = [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / ( 4 - 1 ) s2 = [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 3 s2 = [ 9 + 1 + 1 + 9 ] / 3 = (20 / 3) = 6.67

Problem 6A population consists of four observations: {1, 3, 5, 7}. What is the variance?(A) 2.24 (B) 2.58 (C) 5.00 (D) 6.67 (E) None of the above

SolutionThe correct answer is (D). First, we need to compute the population mean.μ = ( 1 + 3 + 5 + 7 ) / 4 = 4Then we plug all of the known values into formula for the variance of a population, as shown below:σ2 = Σ ( Xi - μ )2 / N σ2 = [ ( 1 - 4 )2 + ( 3 - 4 )2 + ( 5 - 4 )2 + ( 7 - 4 )2 ] / 4 σ2 = [ ( -3 )2 + ( -1 )2 + ( 1 )2 + ( 3 )2 ] / 4 σ2 = [ 9 + 1 + 1 + 9 ] / 4 = 20 / 4 = 5

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