How to find mean - ISEE Upper Level Quantitative Reasoning

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Question

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

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Answer

The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.

The mean is the sum of the elements divided by the number of them. This is

$\frac{67 +68+ 70+70+ 70+ 72+ 74+ 74+ 79}{9} \approx 71.6$

The midrange is the mean of the least and greatest elements, This is $\frac{67+79}{2} =73$

The midrange is the greatest of the four.

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Question

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

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Answer

The mean of the data set is the sum of the elements divided by 5:

$\frac{78 + 92 + 80 + 79 + 74 }{5} = \frac{403 }{5} = 80.6$

The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79.

The mean exceeds the median by

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Question

Consider the following set of numbers:

$\left \{ 20,35,7,12,73,12,18,31\right \}$

Quantity A: Median of the set

Quantity B: Mean of the set

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Answer

The median of the set of numbers is determined by arranging the numbers in numerical order and finding the middle number. In this case there are two middle numbers, and , so we find the average of those numbers, which gives us .

The mean is found by dividing the sum of elements by the number of elements in the set:

$\frac{20+35+7+12+73+12+18+31}{8}=26$

Quantity B is larger.

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Question

Eight judges scored Donna's gymnastics routine as follows:

$\left \{ 8, 6, 7, 7, 9, 10, 5, 9\right \}$

The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain.

What is Donna's score, to the nearest tenth?

Tap to reveal answer

Answer

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

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Question

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Tap to reveal answer

Answer

The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.

The mean is the sum of the elements divided by the number of them. This is

$\frac{67 +68+ 70+70+ 70+ 72+ 74+ 74+ 79}{9} \approx 71.6$

The midrange is the mean of the least and greatest elements, This is $\frac{67+79}{2} =73$

The midrange is the greatest of the four.

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Question

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

Tap to reveal answer

Answer

The mean of the data set is the sum of the elements divided by 5:

$\frac{78 + 92 + 80 + 79 + 74 }{5} = \frac{403 }{5} = 80.6$

The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79.

The mean exceeds the median by

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Question

Consider the following set of numbers:

$\left \{ 20,35,7,12,73,12,18,31\right \}$

Quantity A: Median of the set

Quantity B: Mean of the set

Tap to reveal answer

Answer

The median of the set of numbers is determined by arranging the numbers in numerical order and finding the middle number. In this case there are two middle numbers, and , so we find the average of those numbers, which gives us .

The mean is found by dividing the sum of elements by the number of elements in the set:

$\frac{20+35+7+12+73+12+18+31}{8}=26$

Quantity B is larger.

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Question

Eight judges scored Donna's gymnastics routine as follows:

$\left \{ 8, 6, 7, 7, 9, 10, 5, 9\right \}$

The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain.

What is Donna's score, to the nearest tenth?

Tap to reveal answer

Answer

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

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Question

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Tap to reveal answer

Answer

The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.

The mean is the sum of the elements divided by the number of them. This is

$\frac{67 +68+ 70+70+ 70+ 72+ 74+ 74+ 79}{9} \approx 71.6$

The midrange is the mean of the least and greatest elements, This is $\frac{67+79}{2} =73$

The midrange is the greatest of the four.

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Question

Which is the greater quantity?

(a) The mean of the data set: $\left \{ 1, 0.1, 0.01, 0.001, 0.0001\right \}$

(b) $\frac{2}{9}$

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Answer

(a) The mean of the data set is the sum of its elements divided by :

$\left ( 1+0.1+0.01+0.001+ 0.0001 \right ) \div 5 = 1.1111 \div 5 = 0.22222$

(b) $\frac{2}{9} = 2 \div 9 = 0.222222...$

(b) is greater.

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Question

Eight judges scored Geri's gymnastics routine as follows:

$\left \{ 9, 9, 7, 8, 7, 8, 5, 9\right \}$

The lowest score is disregarded to account for possible bias on the part of the judges. Geri's score is the mean of the scores that remain.

What is Geri's score, to the nearest tenth?

Tap to reveal answer

Answer

The lowest score is 5, so this is disregarded; the remaining seven are averaged by dividing their sum by seven:

$\left ( 9+ 9+ 7+8+ 7+8 +9 \right ) \div 7 = 57 \div 7 \approx 8.1$

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Question

Consider the following set of scores from a physics test. Give the mean of the scores.

$\left \{ 75,73,84,81,68,95,77 \right \}$

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Answer

The mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}=\frac{75+73+84+81+68+95+77}{7}=\frac{553}{7}=79$

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Question

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Tap to reveal answer

Answer

The median of the set is the fifth-highest value, which is 70; this is also the mode, being the most commonly occurring element.

The mean is the sum of the elements divided by the number of them. This is

$\frac{67 +68+ 70+70+ 70+ 72+ 74+ 74+ 79}{9} \approx 71.6$

The midrange is the mean of the least and greatest elements, This is $\frac{67+79}{2} =73$

The midrange is the greatest of the four.

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Question

Which is the greater quantity?

(a) The mean of the data set: $\left \{ 1, 0.1, 0.01, 0.001, 0.0001\right \}$

(b) $\frac{2}{9}$

Tap to reveal answer

Answer

(a) The mean of the data set is the sum of its elements divided by :

$\left ( 1+0.1+0.01+0.001+ 0.0001 \right ) \div 5 = 1.1111 \div 5 = 0.22222$

(b) $\frac{2}{9} = 2 \div 9 = 0.222222...$

(b) is greater.

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Question

Which is greater:

The mean of the data set $\left \{ 3,2,1,\frac{1}{2},\frac{1}{3} \right \}$

$(b)\ 1$

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Answer

Mean of a data set $x_{1},x_{2}, ..., x_{n}$ is the sum of the data set values divided by the number of data:

$Mean=\frac{x_{1}+x_{2}+...+x_{n}}{n}$

So we have:

$Mean = \frac{3+2+1+\frac{1}{2}+\frac{1}{3}}{5}=\frac{\frac{18+12+6+3+2}{6}}{5}=\frac{41}{30}=1\tfrac{11}{30}$

So the mean of the data set is greater than

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Question

Eight judges scored Donna's gymnastics routine as follows:

$\left \{ 8, 6, 7, 7, 9, 10, 5, 9\right \}$

The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain.

What is Donna's score, to the nearest tenth?

Tap to reveal answer

Answer

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

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Question

Consider the following set of scores from a physics test. Give the mean of the scores.

$\left \{ 75,73,84,81,68,95,77 \right \}$

Tap to reveal answer

Answer

The mean of the scores can be calculated as:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}$

Where:

$\bar{x}$ is the mean of a data set, $\sum_{}^{}$ indicates the sum of the data values $x_{i}$ , and is the number of data values. So we can write:

$\bar{x}=\frac{1}{n}\sum_{i=1}^n{x_{i}=\frac{75+73+84+81+68+95+77}{7}=\frac{553}{7}=79$

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Question

What is the mean of the first ten prime numbers?

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Answer

The first ten primes form the data set:

$\left \{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29\right \}$

Add them, and divide by :

$\left ( 2 + 3+5+ 7+ 11+13+17+19+ 23+29 \right )\div 10 = 129\div 10= 12.9$

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Question

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

Tap to reveal answer

Answer

The mean of the data set is the sum of the elements divided by 5:

$\frac{78 + 92 + 80 + 79 + 74 }{5} = \frac{403 }{5} = 80.6$

The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79.

The mean exceeds the median by

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Question

Which is greater:

The mean of the data set $\left \{ 3,2,1,\frac{1}{2},\frac{1}{3} \right \}$

$(b)\ 1$

Tap to reveal answer

Answer

Mean of a data set $x_{1},x_{2}, ..., x_{n}$ is the sum of the data set values divided by the number of data:

$Mean=\frac{x_{1}+x_{2}+...+x_{n}}{n}$

So we have:

$Mean = \frac{3+2+1+\frac{1}{2}+\frac{1}{3}}{5}=\frac{\frac{18+12+6+3+2}{6}}{5}=\frac{41}{30}=1\tfrac{11}{30}$

So the mean of the data set is greater than

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How to find mean - ISEE Upper Level Quantitative Reasoning

Consider the following data set: Which of these numbers is greater than the others?

Which is true about the mean and the median of the following data set:

The mean of the data set is the sum of the elements divided by 5: The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79. The mean exceeds the median by

Consider the following set of numbers: Quantity A: Median of the set Quantity B: Mean of the set

Eight judges scored Donna's gymnastics routine as follows: The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain. What is Donna's score, to the nearest tenth?

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

Consider the following data set: Which of these numbers is greater than the others?

Which is true about the mean and the median of the following data set:

The mean of the data set is the sum of the elements divided by 5: The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79. The mean exceeds the median by

Consider the following set of numbers: Quantity A: Median of the set Quantity B: Mean of the set

Eight judges scored Donna's gymnastics routine as follows: The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain. What is Donna's score, to the nearest tenth?

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

Consider the following data set: Which of these numbers is greater than the others?

Which is the greater quantity? (a) The mean of the data set: (b)

(a) The mean of the data set is the sum of its elements divided by : (b) (b) is greater.

Eight judges scored Geri's gymnastics routine as follows: The lowest score is disregarded to account for possible bias on the part of the judges. Geri's score is the mean of the scores that remain. What is Geri's score, to the nearest tenth?

The lowest score is 5, so this is disregarded; the remaining seven are averaged by dividing their sum by seven:

Consider the following set of scores from a physics test. Give the mean of the scores.

The mean of the scores can be calculated as: Where: is the mean of a data set, indicates the sum of the data values , and is the number of data values. So we can write:

Consider the following data set: Which of these numbers is greater than the others?

Which is the greater quantity? (a) The mean of the data set: (b)

(a) The mean of the data set is the sum of its elements divided by : (b) (b) is greater.

Which is greater: The mean of the data set

Mean of a data set is the sum of the data set values divided by the number of data: So we have: So the mean of the data set is greater than

Eight judges scored Donna's gymnastics routine as follows: The highest and lowest scores are disregarded to account for possible bias on the part of the judges. Donna's score is the mean of the scores that remain. What is Donna's score, to the nearest tenth?

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

Consider the following set of scores from a physics test. Give the mean of the scores.

The mean of the scores can be calculated as: Where: is the mean of a data set, indicates the sum of the data values , and is the number of data values. So we can write:

What is the mean of the first ten prime numbers?

The first ten primes form the data set: Add them, and divide by :

Which is true about the mean and the median of the following data set:

The mean of the data set is the sum of the elements divided by 5: The median of the data set is the middle value when the values are arranged in ascending order, which here is the third-highest value. This is 79. The mean exceeds the median by

Which is greater: The mean of the data set

Mean of a data set is the sum of the data set values divided by the number of data: So we have: So the mean of the data set is greater than

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

Consider the following set of numbers:

$\left \{ 20,35,7,12,73,12,18,31\right \}$

Quantity A: Median of the set

Quantity B: Mean of the set

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

Consider the following set of numbers:

$\left \{ 20,35,7,12,73,12,18,31\right \}$

Quantity A: Median of the set

Quantity B: Mean of the set

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Which is the greater quantity?

(a) The mean of the data set: $\left \{ 1, 0.1, 0.01, 0.001, 0.0001\right \}$

(b) $\frac{2}{9}$

(a) The mean of the data set is the sum of its elements divided by :

$\left ( 1+0.1+0.01+0.001+ 0.0001 \right ) \div 5 = 1.1111 \div 5 = 0.22222$

(b) $\frac{2}{9} = 2 \div 9 = 0.222222...$

(b) is greater.

Eight judges scored Geri's gymnastics routine as follows:

$\left \{ 9, 9, 7, 8, 7, 8, 5, 9\right \}$

The lowest score is disregarded to account for possible bias on the part of the judges. Geri's score is the mean of the scores that remain.

What is Geri's score, to the nearest tenth?

The lowest score is 5, so this is disregarded; the remaining seven are averaged by dividing their sum by seven:

$\left ( 9+ 9+ 7+8+ 7+8 +9 \right ) \div 7 = 57 \div 7 \approx 8.1$

Consider the following set of scores from a physics test. Give the mean of the scores.

$\left \{ 75,73,84,81,68,95,77 \right \}$

Consider the following data set:

$\left \{ 67, 68, 70,70, 70, 72, 74, 74, 79\right \}$

Which of these numbers is greater than the others?

Which is the greater quantity?

(a) The mean of the data set: $\left \{ 1, 0.1, 0.01, 0.001, 0.0001\right \}$

(b) $\frac{2}{9}$

(a) The mean of the data set is the sum of its elements divided by :

$\left ( 1+0.1+0.01+0.001+ 0.0001 \right ) \div 5 = 1.1111 \div 5 = 0.22222$

(b) $\frac{2}{9} = 2 \div 9 = 0.222222...$

(b) is greater.

Which is greater:

The mean of the data set $\left \{ 3,2,1,\frac{1}{2},\frac{1}{3} \right \}$

$(b)\ 1$

The highest and lowest scores are 10 and 5, so these are disregarded; the remaining six are averaged by dividing their sum by six:

$\left ( 8+ 6+ 7+7+ 9 + 9 \right ) \div 6 = 46 \div 6 \approx 7.7$

Consider the following set of scores from a physics test. Give the mean of the scores.

$\left \{ 75,73,84,81,68,95,77 \right \}$

The first ten primes form the data set:

$\left \{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29\right \}$

Add them, and divide by :

$\left ( 2 + 3+5+ 7+ 11+13+17+19+ 23+29 \right )\div 10 = 129\div 10= 12.9$

Which is true about the mean and the median of the following data set:

$\left \{ 78, 92, 80, 79, 74\right \}$

Which is greater:

The mean of the data set $\left \{ 3,2,1,\frac{1}{2},\frac{1}{3} \right \}$

$(b)\ 1$