The mean and the standard deviation for IQ.

Question

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Answer 1

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

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Answer 2

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

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Answer 3

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

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Answer 4

Question

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

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Answer 5

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

Answer 6

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

Answer 7

Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

Answer 8

(a)

Expert Solution

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

x¯=∑i=1nxiN

where N is the total number of observations and ‘i’ varies from 1 to 78.

x¯=111+107+100+⋯+10678=849678=108.9231

Standard deviation for the IQ scores can be calculated as:

s=(∑ixi−x¯)2n−1

Substitute the values in the formula above:

s=(∑ixi−x¯)2n−1=178−1[(111−108.9231)2+(107−108.9231)2+⋯+(106−108.9231)2]=13.17097

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

Answer 13

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

Answer 14

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

Answer 15

(b)

Expert Solution

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 110, and the (n2+1)th is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

Answer 20

(c)

To determine

To find: Mean and median for GPA data.

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.

Answer 21

(c)

To determine

To find: Mean and median for GPA data.

Answer 22

(c)

Expert Solution

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

x¯=∑i=1nxiN

x¯=7.94+8.292+4.643+⋯+6.93878=580.8378=7.4465

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of (n2)th and (n2+1)th observations.

Step 4: The (n2)th observation is the 39^th observation, which is 7.825, and the (n2+1)th is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.