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In mathematics and statistics, the mean is one of the important concepts that can be understood easily without much prior knowledge. Different types of the mean are defined based on the data provided. In this article, you will get questions and solutions for finding the mean of a given data set and grouped data (frequency distribution). These questions will help the students of Classes 9 and 10 to get enough practice for this concept.
What is Mean in Statistics?
Mean is one of the measures of central tendency in statistics. The mean is the average of the given data set, which means it can be calculated by dividing the sum of the given data values by the total number of data values.
Mean for ungrouped data:
Mean (x̄) = ∑x i /n
x i = x 1 , x 2 , x 3 ,…, x n such that i = 1, 2, 3,…n.
Number of observations = n
Mean for grouped data:
Mean (x̄) = ∑f i x i / ∑f i
Here, f i ’s are frequencies of x i ’s.
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1. Calculate the mean from the data showing marks of students in a class in a test: 40, 50, 55, 78, 58.
Given marks:
40, 50, 55, 78, 58
Here, the number of data values = 5
We know that:
Mean = Sum of data values/Total number of data values
= (40 + 50 + 55 + 78 + 58)/5
Therefore, the mean for the given data is 56.2.
2. A class consists of 50 students, out of which 30 are girls. The mean of marks scored by girls in a test is 73 (out of 100), and that of boys is 71. Determine the mean score of the whole class.
Total number of students in a class = 50
Number of girls in the class = 30
Number of boys in the class = 50 – 30 = 20
Mean marks scored by girls = 73
Mean marks scored by boys = 71
Thus, the total marks scored by girls = 73 × 30 = 2190
Also, the total marks scored by boys = 71 × 20 = 1420
Mean score of the class = (Total marks scored by girls and boys)/Total number of students
= (2190 + 1420)/50
3. The mean of the following distribution is 50.
|
|
10 | 17 |
30 | 5a + 3 |
50 | 32 |
70 | 7a – 11 |
90 | 19 |
Find the value of a and hence the frequencies of 30 and 70.
|
|
|
10 | 17 | 170 |
30 | 5a + 3 | 150a + 90 |
50 | 32 | 1600 |
70 | 7a – 11 | 490a – 770 |
90 | 19 | 1710 |
∑f = 12a + 60 | ∑fx = 640a + 2800 |
We know that,
Mean for grouped data (x̄) = ∑f i x i /n
(640a + 2800)/ (12a + 60) = 50 {given that mean = 50}
640a + 2800 = 50(12a + 60)
640a + 2800 = 600a + 3000
640a – 600a = 3000 – 2800
Therefore, the frequency for 30 = 5a + 3 = 5(5) + 3 = 28
And the frequency for 70 = 7a – 11 = 7(5) – 11 = 24
4. Find the mean salary of 60 workers of a factory from the following table:
|
|
3000 | 16 |
4000 | 12 |
5000 | 10 |
6000 | 8 |
7000 | 6 |
8000 | 4 |
9000 | 3 |
10000 | 1 |
Total | 60 |
) | ) | x |
3000 | 16 | 48000 |
4000 | 12 | 48000 |
5000 | 10 | 50000 |
6000 | 8 | 48000 |
7000 | 6 | 42000 |
8000 | 4 | 32000 |
9000 | 3 | 27000 |
10000 | 1 | 10000 |
Total | Σf = 60 | Σf x = 305000 |
Mean = (305000)/60 = 5083.33.
Therefore, the mean salary = Rs. 5083.33
5. A total of 25 patients admitted to a hospital are tested for levels of blood sugar, (mg/dl) and the results obtained were as follows:
87, 71, 83, 67, 85, 77, 69, 76, 65, 85, 85, 54, 70, 68, 80, 73, 78, 68, 85, 73, 81, 78, 81, 77, 75
Find the mean (mg/dl) of the above data.
Sum of data values = 87 + 71 + 83 + 67 + 85 + 77 + 69 + 76 + 65 + 85 + 85 + 54 + 70 + 68 + 80 + 73 + 78 + 68 + 85 + 73 + 81 + 78 + 81 + 77 + 75
Mean = 1891/25
6. Calculate the mean for the following distribution.
Class interval | 10 – 25 | 25 – 40 | 40 – 55 | 55 – 70 | 70 – 85 | 85 – 100 |
Number of students | 2 | 3 | 7 | 6 | 6 | 6 |
CI | Number of students (f ) | Classmarks (x ) | d = x – a | f d |
10 – 25 | 2 | 17.5 | -30 | -60 |
25 – 40 | 3 | 32.5 | -15 | -45 |
40 – 55 | 7 | 47.5 = a | 0 | 0 |
55 – 70 | 6 | 62.5 | 15 | 90 |
70 – 85 | 6 | 77.5 | 30 | 180 |
85 – 100 | 6 | 92.5 | 45 | 270 |
∑f = 30 | ∑f d = 435 |
Mean = a + (∑f i d i /∑f i )
= 47.5 + (435/30)
= 47.5 + 14.5
7. The distribution below shows the number of wickets taken by bowlers in one-day cricket matches. Find the mean number of wickets by choosing a suitable method. What does the mean signify?
Number of wickets | 20 – 60 | 60 – 100 | 100 – 150 | 150 – 250 | 250 – 350 | 350 – 450 |
Number of bowlers | 7 | 5 | 16 | 12 | 2 | 3 |
The given distribution has unequal class heights. So, we can use the direct method for calculating the mean.
Number of wickets | Number of bowlers (f ) | Classmark (x ) | f x |
20 – 60 | 7 | 40 | 280 |
60 – 100 | 5 | 80 | 400 |
100 – 150 | 16 | 125 | 2000 |
150 – 250 | 12 | 200 | 2400 |
250 – 350 | 2 | 300 | 600 |
350 – 450 | 3 | 400 | 1200 |
∑f = 45 | ∑f x = 6880 |
Mean = ∑f i x i / ∑f i
Here the mean signifies that, on average, 45 bowlers take 152.89 wickets.
8. A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Number of plants | 0 – 2 | 2 – 4 | 4 – 6 | 6 – 8 | 8 – 10 | 10 – 12 | 12 – 14 |
Number of houses | 1 | 2 | 1 | 5 | 6 | 2 | 3 |
Which method did you use for finding the mean, and why?
Number of plants | Number of houses (fi) | Classmark (xi) | fixi |
0 – 2 | 1 | 1 | 1 |
2 – 4 | 2 | 3 | 6 |
4 – 6 | 1 | 5 | 5 |
6 – 8 | 5 | 7 | 35 |
8 – 10 | 6 | 9 | 54 |
10 – 12 | 2 | 11 | 22 |
12 – 14 | 3 | 13 | 39 |
∑f = 20 | ∑f x = 162 |
Here, the values of fi and xi are small. Thus, we can use the direct method to calculate the mean.
So, Mean = ∑f i x i / ∑f i
Therefore, the mean number of plants = 8.1
9. The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs 18. Find the missing frequency x.
Daily pocket allowance (in Rs.) | 11 – 13 | 13 – 15 | 15 – 17 | 17 – 19 | 19 – 21 | 21 – 23 | 23 – 25 |
Number of children | 7 | 6 | 9 | 13 | x | 5 | 4 |
Daily pocket allowance (in Rs.) | Number of children (f ) | Classmark (x ) | d = x – a | f d |
11 – 13 | 7 | 12 | -6 | -42 |
13 – 15 | 6 | 14 | -4 | -24 |
15 – 17 | 9 | 16 | -2 | -18 |
17 – 19 | 13 | 18 = a | 0 | 0 |
19 – 21 | x | 20 | 2 | 2x |
21 – 23 | 5 | 22 | 4 | 20 |
23 – 25 | 4 | 24 | 6 | 24 |
∑f = 44 + x | ∑f d = 2x – 40 |
Mean pocket allowance = a + (∑f i d i / ∑f i )
18 + [(2x – 40)/(44 + x)] = 18 [from the given]
(2x – 40)/(44 + x) = 18 – 18
(2x – 40)/(44 + x) = 0
2x – 40 = 0
Hence, the missing frequency is 20.
10. The following is the cumulative frequency distribution (of less than type) of 1000 persons, each one of age 20 years and above. Determine the mean age.
Age below (in yrs) | 30 | 40 | 50 | 60 | 70 | 80 |
Number of persons | 100 | 220 | 350 | 750 | 950 | 1000 |
For the given frequency distribution, we need to write the class intervals and frequencies and then proceed with the necessary calculations for estimating the mean age.
Class interval | Frequency (f ) | Classmark (x ) | u = (x – a)/h = (x – 55)/10 | f u |
20 – 30 | 100 | 25 | -3 | -300 |
30 – 40 | 120 | 35 | -2 | -240 |
40 – 50 | 130 | 45 | -1 | -130 |
50 – 60 | 400 | 55 = a | 0 | 0 |
60 – 70 | 200 | 65 | 1 | 200 |
70 – 80 | 50 | 75 | 2 | 100 |
∑f = 1000 | ∑f u = -370 |
Mean = a + h(∑f i u i /∑f i )
= 55 + 10(-370/1000)
= 55 – 3.7
Therefore, the mean age is 51.3 years.
Find the mean for the data.
Workers | A | B | C | D | E | F | G | H | I | J |
Daily income (in Rs) | 120 | 150 | 180 | 200 | 250 | 300 | 220 | 350 | 370 | 260 |
Area of land (in hectares) | 1 – 3 | 3 – 5 | 5 – 7 | 7 – 9 | 9 – 11 | 11 – 13 |
Number of families | 20 | 45 | 80 | 55 | 40 | 12 |
Find the mean agricultural holdings of the village.
CI | 14 – 15 | 16 – 17 | 18 – 20 | 21 – 24 | 25 – 29 | 30 – 34 | 35 – 39 |
Frequency | 60 | 140 | 150 | 110 | 110 | 100 | 90 |
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Problems Based on Average
Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.
How to solve average word problems?
To solve various problems we need to follow the uses of the formula for calculating arithmetic mean.
Average = (Sums of the observations)/(Number of observations)
Worked-out problems based on average:
1. The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55. Find the weight of the seventh boy.
Mean weight of 7 boys = 56 kg.
Total weight of 7 boys = (56 × 7) kg = 392 kg.
Total weight of 6 boys = (52 + 57 + 55 + 60 + 59 + 55) kg
Weight of the 7th boy = (total weight of 7 boys) - (total weight of 6 boys)
= (392 - 338) kg
Hence, the weight of the seventh boy is 54 kg.
2 . A cricketer has a mean score of 58 runs in nine innings. Find out how many runs are to be scored by him in the tenth innings to raise the mean score to 61.
Mean score of 9 innings = 58 runs.
Total score of 9 innings = (58 x 9) runs = 522 runs.
Required mean score of 10 innings = 61 runs.
Required total score of 10 innings = (61 x 10) runs = 610 runs.
Number of runs to be scored in the 10th innings
= (total score of 10 innings) - (total score of 9 innings)
= (610 -522) = 88.
Hence, the number of runs to be scored in the 10th innings = 88.
3. The mean of five numbers is 28. If one of the numbers is excluded, the mean gets reduced by 2. Find the excluded number.
Mean of 5 numbers = 28.
Sum of these 5 numbers = (28 x 5) = 140.
Mean of the remaining 4 numbers = (28 - 2) =26.
Sum of these remaining 4 numbers = (26 × 4) = 104.
Excluded number
= (sum of the given 5 numbers) - (sum of the remaining 4 numbers)
= (140 - 104)
= 36. Hence, the excluded number is 36.
4 . The mean weight of a class of 35 students is 45 kg. If the weight of the teacher be included, the mean weight increases by 500 g. Find the weight of the teacher.
Mean weight of 35 students = 45 kg.
Total weight of 35 students = (45 × 35) kg = 1575 kg.
Mean weight of 35 students and the teacher (45 + 0.5) kg = 45.5 kg.
Total weight of 35 students and the teacher = (45.5 × 36) kg = 1638 kg.
Weight of the teacher = (1638 - 1575) kg = 63 kg.
Hence, the weight of the teacher is 63 kg.
5. The average height of 30 boys was calculated to be 150 cm. It was detected later that one value of 165 cm was wrongly copied as 135 cm for the computation of the mean. Find the correct mean.
Calculated average height of 30 boys = 150 cm.
Incorrect sum of the heights of 30 boys
= (150 × 30)cm
Correct sum of the heights of 30 boys
= (incorrect sum) - (wrongly copied item) + (actual item)
= (4500 - 135 + 165) cm
Correct mean = correct sum/number of boys
= (4530/30) cm
Hence, the correct mean height is 151 cm.
6. The mean of 16 items was found to be 30. On rechecking, it was found that two items were wrongly taken as 22 and 18 instead of 32 and 28 respectively. Find the correct mean.
Calculated mean of 16 items = 30.
Incorrect sum of these 16 items = (30 × 16) = 480.
Correct sum of these 16 items
= (incorrect sum) - (sum of incorrect items) + (sum of actual items)
= [480 - (22 + 18) + (32 + 28)]
Therefore, correct mean = 500/16 = 31.25.
Hence, the correct mean is 31.25.
7. The mean of 25 observations is 36. If the mean of the first observations is 32 and that of the last 13 observations is 39, find the 13th observation.
Mean of the first 13 observations = 32.
Sum of the first 13 observations = (32 × 13) = 416.
Mean of the last 13 observations = 39.
Sum of the last 13 observations = (39 × 13) = 507.
Mean of 25 observations = 36.
Sum of all the 25 observations = (36 × 25) = 900.
Therefore, the 13th observation = (416 + 507 - 900) = 23.
Hence, the 13th observation is 23.
8. The aggregate monthly expenditure of a family was $ 6240 during the first 3 months, $ 6780 during the next 4 months and $ 7236 during the last 5 months of a year. If the total saving during the year is $ 7080, find the average monthly income of the family.
Total expenditure during the year
= $[6240 × 3 + 6780 × 4 + 7236 × 5]
= $ [18720 + 27120 + 36180]
Total income during the year = $ (82020 + 7080) = $ 89100.
Average monthly income = (89100/12) = $7425.
Hence, the average monthly income of the family is $ 7425.
Arithmetic Mean
Word Problems on Arithmetic Mean
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These lessons help students to learn how to solve word problems involving average.
Related Pages Average Problems Average Speed Problems More Lessons for Algebra Math Worksheets
There are three main types of average problems commonly encountered in school algebra: Average (Arithmetic Mean), Weighted Average and Average Speed .
The following diagram shows the average (arithmetic mean) formula. Scroll down the page for more examples and solutions on how to use formula to solve average word problems.
Example: The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, the average of the remaining numbers is 15. What is the number that was removed?
Step 1: The removed number could be obtained by the difference between the sum of original 6 numbers and the sum of remaining 5 numbers i.e.
Number removed = sum of original 6 numbers – sum of remaining 5 numbers
Step 2: Using the formula
Sum of Terms = Average × Number of Terms
Sum of original 6 numbers = 20 × 6 = 120 Sum of remaining 5 numbers = 15 × 5 = 75
Step 3: Using the formula from step 1
120 – 75 = 45
Answer: The number removed is 45.
Examples of average word problems
How to solve word problems that involve finding the average of a group of numbers?
Word Problems with Averages
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Problem 1 :
John studies for 4 hours, 5 hours and 3 hours respectively on three consecutive days. How many hours does he study daily on an average?
The average study time of John
= Total number of study hours / Number of days for which he studied
= (4 + 5 + 3) / 3
= 12 / 3
= 4 hours
Thus, we can say that John studies for 4 hours daily on an average.
Problem 2 :
A batsman scored the following number of runs in six innings:
36, 35, 50, 46, 60, 55
Calculate the mean runs scored by him in an inning.
To find the mean, we find the sum of all the observations and divide it by the number of observations.
Mean = Total runs / Number of innings
= (36 + 35 + 50 + 46 + 60 + 55) / 6
= 47
Thus, the mean runs scored in an inning are 47.
Problem 3 :
The ages in years of 10 teachers of a school are:
32, 41, 28, 54, 35, 26, 23, 33, 38, 40
What is the mean age of these teachers?
Mean age of the teachers
= Sum of age of teachers / Number of teachers
= (23 + 26 + 28 + 32 + 33 + 35 + 38 + 40 + 41 + 54) /10
= 350 / 10
= 35 years
Problem 4 :
Following table shows the points of each player scored in four games:
Player | Game 1 | Game 2 | Game 3 | Game 4 |
A B C | 14 0 8 | 16 8 11 | 10 6 Did not play | 10 4 13 |
Now answer the following questions:
(i) Find the mean to determine A’s average number of points scored per game.
(ii) To find the mean number of points per game for C, would you divide the total points by 3 or by 4? Why?
(iii) B played in all the four games. How would you find the mean?
(iv) Who is the best performer?
(i) Mean score of A = (14 + 16 + 10 + 10) / 4
= 12.5
Mean score of A per game is 12.5
(ii) To find the mean number of points per game for C, we have to divide the total points by 3.
Because he didn't participate in game 3. Total number of games he played is 3.
(iii) Mean score of B = (0 + 8 + 6 + 4) / 4
= 18/4
= 4.5
Mean score of B per game is 4.5
(iv) To choose the best performer, we have to find the mean score of each player.
Mean score of C = (8 + 11 + 13) / 3
= 32/3
= 10.6
Mean score of C per game is 10.6
Hence C is the best performer.
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The arithmetic mean of a set of numbers (or variables) is the sum of all the numbers, divided by the number of numbers - the average of the set. If we let denote Arithmetic Mean, . is the arithmetic mean of the numbers .. For example, if I wanted to find the average of the numbers 3, 1, 4, 1, and 5, I would compute: Arithmetic means show up frequently in contest problems, often in the AM-GM ...
Arithmetic mean is calculated by adding all the numbers in a given set and then dividing by the total number of items within that set. Learn different arithmetic mean formulas used for different types of data. ... Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts. Book a Free Trial Class.
Arithmetic Mean Formula. To find the arithmetic mean of a set of observations, just add them all together and divide the sum by the number of data points. To compute the mean set of observations, use the arithmetic mean formula: Arithmetic mean = Sum of all given values / Total number of values. Also, read: Mean
Follow the explanation to solve the word problems on arithmetic mean (average): 1. The heights of five runners are 160 cm, 137 cm, 149 cm, 153 cm and 161 cm respectively. Find the mean height per runner. Solution: Mean height = Sum of the heights of the runners/number of runners. = (160 + 137 + 149 + 153 + 161)/5 cm.
How to find the arithmetic mean or average? Add up all the numbers. Divide the total by how many there are. Solving Math Problems : Finding the Mean. In math, the mean is found by adding pieces of data together and dividing by the number of pieces of data. Example: Find the mean of 2, 8, 13, 4, 21, 18.
Arithmetic Mean Formula Sum of all of the numbers of a group, when divided by the number of items in that list is known as the Arithmetic Mean or Mean of the group. For example, the mean of the numbers 5, 7, 9 is 4 since 5 + 7 + 9 = 21 and 21 divided by 3 [there are three numbers] is 7.
The mean (also known as the arithmetic mean) is a measure of central tendency because it describes a set of numbers by identifying a central position within the data. ... Example 5: problem solving. The mean of 4 values is 10 . Here are 3 of the values: 6 9 12 . Find the 4 th value.
The arithmetic mean, geometric mean, harmonic mean, and root mean square are all special cases of the power mean. Inequalities and Optimization. There are numerous inequalities that relate different types of means. The most common are part of the RMS-AM-GM-HM inequality chain. This inequality chain is a set of special cases of the Power mean ...
The mean in math, specifically the arithmetic mean, is a type of average calculated by finding the total of the values and dividing the total by the number of values. \text{Mean}=\cfrac{\text{total}}{\text{number of values}} ... Example 5: problem solving. The mean of 4 values is 10. Here are 3 of the values: 6 \quad 9 \quad 12 . Find the 4^{th ...
The arithmetic mean of a set of numbers (or variables) is the sum of all the numbers, divided by the number of numbers - the average of the set. If we let denote Arithmetic Mean, . is the arithmetic mean of the numbers .. For example, if I wanted to find the average of the numbers 3, 1, 4, 1, and 5, I would compute: Arithmetic means show up frequently in contest problems, often in the AM-GM ...
Calculating the mean. The following table shows the number of raisins in a scoop of different brands of raisin bran cereal. Find the mean number of raisins. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing ...
Average (Arithmetic Mean) The average (arithmetic mean) uses the formula: The formula can also be written as Sum of Terms = Average × Number of Terms. Example: The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, the average of the remaining numbers is 15. What is the number that was removed? Solution:
For ungrouped data, we can easily find the arithmetic mean by adding all the given values in a data set and dividing it by a number of values. Mean, x̄ = Sum of all values/Number of values. Example: Find the arithmetic mean of 4, 8, 12, 16, 20. Solution: Given, 4, 8, 12, 16, 20 is the set of values. Sum of values = 4+ 8+12+16+20 = 60.
Mean: The "average" number; found by adding all data points and dividing by the number of data points. Example: The mean of 4 , 1 , and 7 is ( 4 + 1 + 7) / 3 = 12 / 3 = 4 . Median: The middle number; found by ordering all data points and picking out the one in the middle (or if there are two middle numbers, taking the mean of those two numbers).
These average word problems are based on the arithmetic mean usually called average or mean. To get the average, use the formula for average you see here. First, study the example below. A word problem about finding the average.
A. $$0$$ Eliminate some obvious wrong ones B, C, E.Remember to always estimate and have a good rough estimate of where the answers should be which is right around the original average of $$9$$; if you added a new number that was much higher or lower like say $$0$$ or $$1,000$$ the average could never remain $$9$$.
Solution (two solutions - balancing and summing) We know the average is 15. 3, 5, and 7 are, respectively, 12, 10, and 8 below the average. So far then we are -12 - 10 - 8 or 30 below the average. We have to make this up with a and b so on average each is 30/2 = 15 above the average. The average is 15 so each is on average 15+15 = 30, or answer C.
What is Mean in Statistics? Mean is one of the measures of central tendency in statistics. The mean is the average of the given data set, which means it can be calculated by dividing the sum of the given data values by the total number of data values. Mean for ungrouped data: Mean (x̄) = ∑x i /n . x i = x 1, x 2, x 3,…, x n such that i = 1 ...
The following table gives the formulas for average problems: Weighted Average, Mean, and Average Speed. Scroll down the page for examples and solutions. Weighted Average Problems. One type of average problems involves the weighted average - which is the average of two or more terms that do not all have the same number of members.
To solve various problems we need to follow the uses of the formula for calculating arithmetic mean. Average = (Sums of the observations)/ (Number of observations) Worked-out problems based on average: 1. The mean weight of a group of seven boys is 56 kg. The individual weights (in kg) of six of them are 52, 57, 55, 60, 59 and 55.
Try another pair of numbers for example, 1 and 2. The arithmetic mean is 1.5; the geometric mean is \(\sqrt{2} ≈ 1.414\). For both pairs, the geometric mean is smaller than the arithmetic mean. This pattern is general; it is the famous arithmetic-mean-geometric-mean (AM-GM) inequality [18]: \[\begin{aligned}
Average (Arithmetic Mean) The following diagram shows the average (arithmetic mean) formula. Scroll down the page for more examples and solutions on how to use formula to solve average word problems. Example: The average (arithmetic mean) of a list of 6 numbers is 20. If we remove one of the numbers, the average of the remaining numbers is 15.
Problem 2 : A batsman scored the following number of runs in six innings: 36, 35, 50, 46, 60, 55. Calculate the mean runs scored by him in an inning. Solution : To find the mean, we find the sum of all the observations and divide it by the number of observations. Mean = Total runs / Number of innings = (36 + 35 + 50 + 46 + 60 + 55) / 6 = 47