Calculate the median and mode for the frequency distribution given below: Variable 10 - 13 13 - 16 16 - 19 19 - 22 22 - 25 25 - 28 28 - 31 31 - 34 34 - 37 37 - 40 Frequency 8 15 27 51 75 54 36 18 9 7

Variable Frequency f Cumulative Frequency 10 - 13 8 8 13 - 16 15 23 16 - 19 27 50 19 - 22 51 101 22 - 25 75 176 25 - 28 54 230 28 - 31 36 266 31 - 34 18 284 34 - 37 9 293 37 - 40 7 300 Here, N = Σ f = 300 So, N 2 = 300 2 = 150 We Notice That 150 th value lies in class 22 - 25 , Therefore, 22 - 25 is the median class. We know, Median = l + N 2 - C f × h l = the lower limit of median class = 22 , Frequency of the median class = f = 75 , Cumulative frequency of the class preceding the median class = C = 101 , Width of the median class = h = 3 , Sum of all frequencies = N , Median = 22 + 150 - 101 75 × 3 = 22 + 1 . 96 Median = 23 . 96 Here, the highest frequency is 75 , corresponding to this modal class is 22 - 25 . Lower limit of the modal class = l = 22 , Frequency of the modal class = f m = 75 , Frequency of the class preceding the modal class = f m - 1 = 51 , Frequency of the class succeeding the modal class = f m + 1 = 54 , h = 3 , Mode = l + f m - f m - 1 2 f m - f m - 1 - f m + 1 × h = 22 + 75 - 51 2 × 75 - 51 - 54 × 3 = 22 + 24 × 3 45 Mode = 23 . 6 Hence, median = 23 . 96 , Mode = 23 . 6 .

Estimate the median and mode of the following data : Wages ₹ Number of workers Less than 10 3 Less than 20 12 Less than 30 27 Less than 40 57 Less than 50 75 Less than 60 80

Wages Number of workers Cumulative frequency 0 - 10 3 3 10 - 20 9 12 20 - 30 15 27 30 - 40 30 57 40 - 50 18 75 50 - 60 5 80 Here, N = ∑ f = 80 So, N 2 = 80 2 = 40 We notice that 40 th value lies in class 30 - 40 . We know, Median = l + N 2 - C f × h Here, l = the lower limit of median class = 30 , Frequency of the median class = f = 30 , Cumulative frequency of the class preceding the median class = C = 27 , Width of the median class = h = 10 , Sum of all frequencies = N = 80 , Median = 30 + 40 - 27 30 × 10 = 30 + 4 . 33 Median = 34 . 33 Here, highest frequency is 30 , corresponding to this modal class is 30 - 40 . Lower limit of the modal class = l = 30 , Frequency of the modal class = f m = 30 , Frequency of the class preceding the modal class = f m - 1 = 15 , Frequency of the class succeeding the modal class = f m + 1 = 18 , h = 10 , Mode = l + f m - f m - 1 2 f m - f m - 1 - f m + 1 × h Mode = 30 + ( 30 - 15 ) × 10 ( 2 × 30 - 15 - 18 ) = 30 + 15 27 × 10 = 30 + 5 . 55 Mode = 35 . 55

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The median and mean of a moderately asymmetrical frequency distribution are $72$ and $74$ respectively. Estimate the mode.

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Important Questions on Statistics

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11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 8

The mean and mode of a moderately skewed distribution are

90

and

96

respectively. Estimate the median.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 9

Calculate the median and mode for the frequency distribution given below:

Variable	$10 - 13$	$13 - 16$	$16 - 19$	$19 - 22$	$22 - 25$	$25 - 28$	$28 - 31$	$31 - 34$	$34 - 37$	$37 - 40$
Frequency	$8$	$15$	$27$	$51$	$75$	$54$	$36$	$18$	$9$	$7$

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11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 10

Estimate the median and mode of the following data :

Wages $(₹)$	Number of workers
Less than $10$	$3$
Less than $20$	$12$
Less than $30$	$27$
Less than $40$	$57$
Less than $50$	$75$
Less than $60$	$80$

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11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 1

The following table shows the distribution of the heights of a group of workers in a factory:

Height (in $cm$ )	$150 - 155$	$155 - 160$	$160 - 165$	$165 - 170$	$170 - 175$	$175 - 180$	$180 - 185$
Number of workers	$6$	$12$	$18$	$20$	$13$	$8$	$6$

Convert the distribution to more than cumulative frequency distribution and draw its ogive. Hence, obtain the median from the graph.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 2

Attempt this question on a graph paper. The table shows the distribution of the marks obtained by $160$ students in a test:

Marks	$0 - 10$	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$
Number of Students	$12$	$20$	$30$	$38$	$24$	$16$	$12$	$8$

Construct more than type cumulative frequency table and draw its ogive. From the ogive, determine the median marks.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 6

$100$ students in a school have heights as tabulated below:

Height (in $cm$ )	$121 - 130$	$131 - 140$	$141 - 150$	$151 - 160$	$161 - 170$	$171 - 180$
Number of pupils	$12$	$16$	$30$	$20$	$14$	$8$

Draw the ogive for the above data and from it determine the median (use graph paper.)

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 8

Draw an ogive for the following data:

Class	$1 - 100$	$101 - 200$	$201 - 300$	$301 - 400$	$401 - 500$	$501 - 600$	$601 - 700$	$701 - 800$
Frequency	$23$	$25$	$35$	$60$	$45$	$39$	$25$	$16$

Estimate the median from your graph.

HARD

11th ICSE

IMPORTANT

Understanding ISC Mathematics Class 11 Volume 2>Chapter 1 - Statistics>EXERCISE>Q 11

The table shows the frequency distribution of $weights$ of $80$ apples selected at random from a big consignment.

$Weight (g)$	$110 - 119$	$120 - 129$	$130 - 139$	$140 - 149$	$150 - 159$	$160 - 169$	$170 - 179$	$180 - 189$
Frequency	$5$	$7$	$12$	$20$	$16$	$10$	$7$	$3$

Draw both cumulative (less than) and cumulative (more than) ogives on a graph paper and determine the median $weight$ of apples from the point of intersection of these two curves.

The median and mean of a moderately asymmetrical frequency distribution are 72 and 74 respectively. Estimate the mode.

Important Questions on Statistics

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The median and mean of a moderately asymmetrical frequency distribution are $72$ and $74$ respectively. Estimate the mode.