Shown below are number line representation of the medians of two different data sets. Which of the two representations of median could be correct?

Cannot say without knowing the elements of the data sets

100 surnames were randomly picked up from a local telephone directory. The frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows Number of letters 1 - 4 4 - 7 7 - 10 10 - 13 13 - 16 16 - 19 Number of surnames 6 30 40 16 4 4 Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames up to two places after decimal point.

We have to find the mean, the median and the mode of the given data. For the mean we shall use the step-deviation method. The frequency distribution table is given below. Number of letters Number of surnames ( f i ) Cumulative frequency ( c f ) Classmark ( x i ) d i = x i - a u i = x i - a h f i u i 1 - 4 6 6 2 . 5 - 6 - 2 - 12 4 - 7 30 36 5 . 5 - 3 - 1 - 30 7 - 10 40 76 8 . 5 0 0 0 10 - 13 16 92 11 . 5 3 1 16 13 - 16 4 96 14 . 5 6 2 8 16 - 19 4 100 17 . 5 9 3 12 Total ∑ f i = 100 ∑ f i u i = - 6 We know that the mean in the step-deviation method is given by x ¯ = a + ∑ f i u i ∑ f i × h . Here, a = 8 . 5 , h = 3 , ∑ f i u i = - 6 and ∑ f i = 100 . So, the mean x ¯ = a + ∑ f i u i ∑ f i × h = 8 . 5 + - 6 100 × 3 = 8 . 5 - 0 . 18 = 8 . 32 Hence, the mean number of letters in the surnames is 8 . 32 . Now, we know that the median for a grouped data is given as median = l + n 2 - c f f × h , where, l = lower boundary of median class, n = number of observations, c f = cumulative frequency of class preceding the median class, f = frequency of median class, h = class size (size of the median class). Here, n = 100 . So, n 2 = 100 2 = 50 , which belongs to the class 7 - 10 . Hence, the median class is 7 - 10 So, in this case, n = 100 , l = 7 , c f = 36 , f = 40 and h = 10 - 7 = 3 So, the median = l + n 2 - c f f × h = 7 + 50 - 36 40 × 3 = 7 + 1 . 05 = 8 . 05 Hence, the median number of letters in the surnames is 8 . 05 Now, for grouped data, the mode is calculated using the following formula. Mode = l + f 1 - f 0 2 f 1 - f 0 - f 2 × h , where, l = the lower boundary of the modal class, h = size of the modal class interval, f 1 = frequency of the modal class, f 0 = frequency of the class preceding the modal class, f 2 = frequency of the class succeeding the modal class. Here, the modal class is 7 - 10 [The class with the maximum frequency, called the modal class]. So, here, l = 7 , h = 10 - 7 = 3 , f 1 = 40 , f 0 = 30 and f 2 = 16 . Hence, the mode = 7 + 40 - 30 2 × 40 - 30 - 16 × 3 = 7 + 10 80 - 46 × 3 = 7 + 0 . 88 = 7 . 88 . [up to two places after decimal point] So, the modal size of the surnames is 7 . 88 letters. Therefore, the mean, the median and the mode number of letters in the surnames are 8 . 32 , 8 . 05 and 7 . 88 , respectively.

MEDIUM

10th CBSE

IMPORTANT

Earn 100

In a class test, the mean score of the class is $60$ . Half the students of the class scored $80$ marks or above in the test.

Dipti said, "Each of the remaining half of the students would have definitely got $40$ marks or below in the test for the mean to be $60$ marks".

Prove or disprove Dipti's statement with a valid example.

Important Questions on Statistics

MEDIUM

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 4

Bowling strike rate for a bowler is defined as the average number of balls bowled per wicket taken. A bowler has taken $148$ wickets so far with a strike rate of $27$ . In his next match, he bowls $48$ balls and takes $2$ wickets. What is his new strike rate? Show your work.

MEDIUM

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 5

The mean temperature of a certain city for $31$ consecutive days was found to be $35.7 ° C$ . Further, the mean temperature of the first $8$ days was $28.4 ° C$ . The mean temperature of the next $12$ days was $36.4 ° C$ .

Find the mean temperature of the rest of the days. Show your work. (Note: Round the numbers to one decimal point.)

EASY

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 6

A ketchup manufacturing company sells ketchup in three different bottle sizes. Now, it wants to sell only one size in the market. It has data on how the three sizes perform in the market.

Based on which measure of central tendency, should the company fix the size of the ketchup bottle? Justify your answer.

EASY

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 7

A survey was conducted on $80$ gamers on how many games did they play in a day. The data is given below.

Number of games	Number of gamers
$1 - 2$	$20$
$2 - 3$	$24$
$3 - 4$	$10$
$4 - 5$	$12$
$5 - 6$	$8$
$6 - 7$	$4$
$7 - 8$	$2$

Which of the following is the modal class?

EASY

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 8

Shown below are number line representation of the medians of two different data sets.

Question Image

Which of the two representations of median could be correct?

MEDIUM

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 9

In statistics, an outlier is a data point that differs significantly from other observations of data set. If an outlier is included in the following data set, which measure(s) of central tendency would change?

$12, 15, 22, 44, 44, 48, 50, 51$

HARD

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 10

An incomplete distribution is given below:

Variable	$10 - 20$	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$
Frequency	$12$	$30$	_____	$65$	_____	$25$	$18$

You are given that the median value is $46$ and the total number of items is $230$ .

(i) Using the median formula, fill up the missing frequencies.

(ii) Calculate the A.M. of the completed distribution.

HARD

10th CBSE

IMPORTANT

Mathematics Textbook of Competency Based Questions for Class X>Chapter 14 - Statistics>Exercise>Q 11

$100$ surnames were randomly picked up from a local telephone directory.

Question Image

The frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows

Number of letters	$1 - 4$	$4 - 7$	$7 - 10$	$10 - 13$	$13 - 16$	$16 - 19$
Number of surnames	$6$	$30$	$40$	$16$	$4$	$4$

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames up to two places after decimal point.

Important Questions on Statistics

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