Selina Concise Solutions for ICSE Class 6 Mathematics Chapter 34 Mean and Median

Exercise 34(A)

Question 1. Find the mean of :
(i) 7, 10, 4 and 17
(ii) 12, 9, 6, 11 and 17
(iii) 3, 1, 5, 4, 4 and 7
(iv) 7, 5, 0, 3, 0, 6, 0, 9, 1 and 4
(v) 2.1, 4.5, 5.2, 7.1 and 9.3
(vi) 5, 2.4, 6.2, 8.9, 4.1 and 3.4
Answer:
(i) First, find the sum of all the numbers:
Sum = \( 7 + 10 + 4 + 17 = 38 \)
There are 4 numbers in total.
Mean = \( \frac{38}{4} = 9.5 \)

(ii) For the numbers 12, 9, 6, 11 and 17:
The total count of observations is 5.
Sum of these numbers = \( 12 + 9 + 6 + 11 + 17 = 55 \)
Mean = \( \frac{55}{5} = 11 \)

(iii) For the data 3, 1, 5, 4, 4 and 7:
The total number of observations is 6.
Sum of these observations = \( 3 + 1 + 5 + 4 + 4 + 7 = 24 \)
Mean = \( \frac{24}{6} = 4 \)

(iv) For the numbers 7, 5, 0, 3, 0, 6, 0, 9, 1 and 4:
The total number of observations is 10.
Sum of these observations = \( 7 + 5 + 0 + 3 + 0 + 6 + 0 + 9 + 1 + 4 = 35 \)
Mean = \( \frac{35}{10} = 3.5 \)

(v) For the decimal values 2.1, 4.5, 5.2, 7.1 and 9.3:
The count of values is 5.
Sum of the values = \( 2.1 + 4.5 + 5.2 + 7.1 + 9.3 = 28.2 \)
Mean = \( \frac{28.2}{5} = 5.64 \)

(vi) For 5, 2.4, 6.2, 8.9, 4.1 and 3.4:
The count of values is 6.
Sum of the values = \( 5 + 2.4 + 6.2 + 8.9 + 4.1 + 3.4 = 30 \)
Mean = \( \frac{30}{6} = 5 \)

In simple words: To find the mean of a group of numbers, add them all up first. Then, divide that total sum by how many numbers there are.

Exam Tip: Always count the number of data values carefully, especially when there are zeros in the set. Even a zero must be counted as a data value.

Question 2. Find the mean of :
(i) first eight natural numbers
(ii) first six even natural numbers
(iii) first five odd natural numbers
(iv) all prime numbers upto 30
(v) all prime numbers between 20 and 40.
Answer:
(i) The first eight natural numbers are \( 1, 2, 3, 4, 5, 6, 7, 8 \).
Total sum of these values = \( 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 \).
The count of numbers is \( 8 \).
Mean = \( \frac{36}{8} = 4.5 \).

(ii) The first six even natural numbers are \( 2, 4, 6, 8, 10, 12 \).
Total sum of these values = \( 2 + 4 + 6 + 8 + 10 + 12 = 42 \).
The count of numbers is \( 6 \).
Mean = \( \frac{42}{6} = 7 \).

(iii) The first five odd natural numbers are \( 1, 3, 5, 7, 9 \).
Total sum of these values = \( 1 + 3 + 5 + 7 + 9 = 25 \).
The count of numbers is \( 5 \).
Mean = \( \frac{25}{5} = 5 \).

(iv) The prime numbers up to 30 are \( 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \).
Total sum of these values = \( 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129 \).
The count of numbers is \( 10 \).
Mean = \( \frac{129}{10} = 12.9 \).

(v) The prime numbers between 20 and 40 are \( 23, 29, 31, 37 \).
Total sum of these values = \( 23 + 29 + 31 + 37 = 120 \).
The count of numbers is \( 4 \).
Mean = \( \frac{120}{4} = 30 \).

In simple words: First, list down the specific numbers asked in each part. Add them up to find the total sum, and then divide by how many numbers you listed.

Exam Tip: Remember that 1 is neither prime nor composite, and 2 is the only even prime number. Write down the lists carefully to avoid skipping any values.

Question 3. Height (in cm) of 7 boys of a locality are 144 cm, 155 cm, 168 cm, 163 cm, 167 cm, 151 cm and 158 cm. Find their mean height.
Answer:
First, we add up the heights of all seven boys:
Sum of heights = \( 144 + 155 + 168 + 163 + 167 + 151 + 158 = 1106 \) cm.
Total number of boys = \( 7 \).
Mean height = \( \frac{1106}{7} = 158 \) cm.

In simple words: Find the sum of the heights of all the boys. Then divide this total by the number of boys to get the average height.

Exam Tip: Always write the final unit of measurement, which is cm in this case, to avoid losing marks.

Question 4. Find the mean of 35, 44, 31, 57, 38, 29, 26, 36, 41 and 43.
Answer:
First, calculate the sum of all the given numbers:
Sum = \( 35 + 44 + 31 + 57 + 38 + 29 + 26 + 36 + 41 + 43 = 380 \).
Total number of values = \( 10 \).
Mean = \( \frac{380}{10} = 38 \).

In simple words: Add all the ten numbers together to find their sum. Divide this sum by ten to get the mean.

Exam Tip: Group numbers in pairs that add up to multiples of 10 to make addition faster and error-free during exams.

Question 5. The mean of 18, 28, x, 32, 14 and 36 is 23. Find the value of x.
Answer:
The total count of observations is 6.
We are given that the mean is 23.
Using the mean formula:
\( \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \)
\( 23 = \frac{18 + 28 + x + 32 + 14 + 36}{6} \)

\( \implies 23 = \frac{128 + x}{6} \)

\( \implies 23 \times 6 = 128 + x \)

\( \implies 138 = 128 + x \)

\( \implies x = 138 - 128 \)

\( \implies x = 10 \)

In simple words: Add all the numbers including x. Set up an equation with their average, which is 23. Multiply by 6 and solve to find that x is 10.

Exam Tip: Remember to multiply the mean by the total number of items first to find the sum of all the numbers.

Question 6. If the mean of x, x + 2, x + 4, x + 6 and x + 8 is 13, find the value of x.
Answer:
There are 5 terms in total: \( x, x + 2, x + 4, x + 6, \) and \( x + 8 \).
The given mean is 13.
Using the mean formula:
\( 13 = \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8)}{5} \)

\( \implies 13 = \frac{5x + 20}{5} \)

\( \implies 13 \times 5 = 5x + 20 \)

\( \implies 65 = 5x + 20 \)

\( \implies 5x = 65 - 20 \)

\( \implies 5x = 45 \)

\( \implies x = \frac{45}{5} \)

\( \implies x = 9 \)

In simple words: Add up the five algebraic terms to get 5x + 20. Divide this by 5 and make it equal to the given mean of 13. Then, solve the equation to find x.

Exam Tip: Be careful when adding the terms; make sure you count all the x variables and add up the constants separately.

Exercise 34(B)

Question 1. Find the median of
(i) 21, 21, 22, 23, 23, 24, 24, 24, 24, 25 and 25
(ii) 3.2, 4.8, 5.6, 5.6, 7.3, 8.9 and 9.1
(iii) 17, 23, 36, 12, 18, 23, 40 and 20
(iv) 26, 33, 41, 18, 30, 22, 36, 45 and 24
(v) 80, 48, 66, 61, 75, 52, 45 and 70
Answer:
(i) The given data in ascending order is: \( 21, 21, 22, 23, 23, 24, 24, 24, 24, 25, 25 \).
The total number of terms is 11, which is odd.
The median is the middlemost term, which is the 6th term.
Thus, Median = \( 24 \).

(ii) The data is already sorted in ascending order: \( 3.2, 4.8, 5.6, 5.6, 7.3, 8.9, 9.1 \).
The number of terms is 7, which is odd.
The median is the 4th term.
Thus, Median = \( 5.6 \).

(iii) First, arrange the given numbers in ascending order:
\( 12, 17, 18, 20, 23, 23, 36, 40 \).
Here, the number of terms \( n = 8 \), which is an even number.
Median = \( \frac{1}{2} \left[ \left(\frac{n}{2}\right)\text{th term} + \left(\frac{n}{2} + 1\text{)}th term\right) \right] \)

\( \implies \text{Median} = \frac{1}{2} \left[ 4\text{th term} + 5\text{th term} \right] \)

\( \implies \text{Median} = \frac{1}{2} [ 20 + 23 ] \)

\( \implies \text{Median} = \frac{1}{2} \times 43 = 21.5 \)

(iv) Sorting the values in ascending order gives:
\( 18, 22, 24, 26, 30, 33, 36, 41, 45 \).
The count of values \( n = 9 \), which is an odd number.
Median = \( \left(\frac{n + 1}{2}\right)\text{th term} \)

\( \implies \text{Median} = \left(\frac{9 + 1}{2}\right)\text{th term} = 5\text{th term} \)
Looking at our sorted list, the 5th term is 30.
Thus, Median = \( 30 \).

(v) Sorting the numbers in ascending order gives:
\( 45, 48, 52, 61, 66, 70, 75, 80 \).
The number of terms \( n = 8 \), which is an even number.
Median = \( \frac{1}{2} \left[ \left(\frac{n}{2}\right)\text{th term} + \left(\frac{n}{2} + 1\text{)}th term\right) \right] \)

\( \implies \text{Median} = \frac{1}{2} \left[ 4\text{th term} + 5\text{th term} \right] \)

\( \implies \text{Median} = \frac{1}{2} [ 61 + 66 ] \)

\( \implies \text{Median} = \frac{1}{2} \times 127 = 63.5 \)

In simple words: To find the median, first write the list of numbers from smallest to largest. If there is an odd number of values, the median is the one right in the middle. If there is an even number of values, add the two middle numbers together and divide by 2.

Exam Tip: Never calculate the median without first sorting the data in ascending order. This is the most common mistake students make.

Question 2. Find the mean and the median of :
(i) 1,3,4, 5, 9, 9 and 11
(ii) 10,12, 12, 15, 15, 17, 18, 18, 18 and 19
(iii) 2, 4, 5, 8, 10,13 and 14
(iv) 5, 8, 10, 11,13, 16, 19 and 20
(v) 1.2, 1.9, 2.2, 2.6 and 2.9
(vi) 0.5, 5.6, 3.8, 4.9, 2.7 and 4.4
Answer:
(i) The given data is: \( 1, 3, 4, 5, 9, 9, 11 \).
The data is already sorted.
The number of terms \( n = 7 \) (odd).
Median = 4th term = \( 5 \).
Mean = \( \frac{1 + 3 + 4 + 5 + 9 + 9 + 11}{7} = \frac{42}{7} = 6 \).

(ii) The sorted data is: \( 10, 12, 12, 15, 15, 17, 18, 18, 18, 19 \).
The number of terms \( n = 10 \) (even).
Median = \( \frac{1}{2} [ 5\text{th term} + 6\text{th term} ] = \frac{1}{2} [ 15 + 17 ] = \frac{1}{2} [ 32 ] = 16 \).
Mean = \( \frac{10 + 12 + 12 + 15 + 15 + 17 + 18 + 18 + 18 + 19}{10} = \frac{154}{10} = 15.4 \).

(iii) The sorted data is: \( 2, 4, 5, 8, 10, 13, 14 \).
The number of terms \( n = 7 \) (odd).
Median = 4th term = \( 8 \).
Mean = \( \frac{2 + 4 + 5 + 8 + 10 + 13 + 14}{7} = \frac{56}{7} = 8 \).

(iv) The sorted data is: \( 5, 8, 10, 11, 13, 16, 19, 20 \).
The number of terms \( n = 8 \) (even).
Median = \( \frac{1}{2} [ 4\text{th term} + 5\text{th term} ] = \frac{1}{2} [ 11 + 13 ] = \frac{1}{2} [ 24 ] = 12 \).
Mean = \( \frac{5 + 8 + 10 + 11 + 13 + 16 + 19 + 20}{8} = \frac{102}{8} = 12.75 \).

(v) The sorted data is: \( 1.2, 1.9, 2.2, 2.6, 2.9 \).
The number of terms \( n = 5 \) (odd).
Median = 3rd term = \( 2.2 \).
Mean = \( \frac{1.2 + 1.9 + 2.2 + 2.6 + 2.9}{5} = \frac{10.8}{5} = 2.16 \).

(vi) Arranging the data in ascending order: \( 0.5, 2.7, 3.8, 4.4, 4.9, 5.6 \).
The number of terms \( n = 6 \) (even).
Median = \( \frac{1}{2} [ 3\text{rd term} + 4\text{th term} ] = \frac{1}{2} [ 3.8 + 4.4 ] = \frac{1}{2} \times 8.2 = 4.1 \).
Mean = \( \frac{0.5 + 2.7 + 3.8 + 4.4 + 4.9 + 5.6}{6} = \frac{21.9}{6} = 3.65 \).

In simple words: To find both values, start by putting the list in order. The mean is the total sum of all numbers divided by how many there are. The median is the middle value or the average of the two middle values.

Exam Tip: Be double sure to show all the steps for both calculations separately, as examiners allocate separate marks for mean and median formulas.