Statistics

Theory.

The number of times a particular observation occurs in data is called frequency.

Showing data in tabular form with showing frequency of each distribution. This representation is called Frequency distribution table.

(i) As looking in Frequency distribution table

Highest Frequency is 16

And the group of marks having highest frequency is 40-49

∴ Maximum number of students have got marks between 40 to 49.

Lowest Frequency is 1

And the group of marks having lowest frequency is 90-99

∴ Minimum number of students have got marks between 90 to 99.

(ii) The given distribution is in inclusive form . It should be converted in exclusive form

Upper limit of 1^st interval is 39

Lower limit of 2^nd interval is 40

= = = 0.5

Actual upper limit = Stated upper limit + = 39 + 0.5 = 39.5

Actual lower limit = Stated lower limit – = 30 – 0.5 = 29.5

∴ Upper limit of group 30-39 is 29.5

Lower limit is lower most value of group

∴ Lower limit of group 30-39 is 30.5

(iii) Range = highest – lowest

Highest marks in class is 97

Lowest marks in class is 32

Range = 97 – 32 = 65

Theory.

The number of times a particular observation occurs in data is called frequency.

Showing data in tabular form with showing frequency of each distribution. This representation is called Frequency distribution table.

(i) Third interval is 30-40

The frequency of the Third interval is 10

(ii) Class size = upper limit – lower limit

In class interval 30-40

Upper limit of interval is 40

Lower limit of interval is 30

Class size = 40 – 30 = 10

Midpoint =

= = = 35

(iii) Range = highest – lowest

Highest Score is 64

Lowest Score is 12

Range = 64 – 12 = 52

The given distribution is in inclusive form. It should be converted in exclusive form

Upper limit of 1^st interval is 19

Lower limit of 2^nd interval is 20

= = = 0.5

Actual upper limit = Stated upper limit +

Actual lower limit = Stated lower limit –

∴ Frequency distribution table

Theory.

Average =

Solution.

Average =

Sum of runs scored by 10 batsmen

= 23 + 54 + 8 + 94 + 60 + 18 + 29 + 44 + 5 + 86

= 421

There are 10 batsmen

Average =

∴ Average = = 42.1 runs

To find the mean let us prepare frequency distribution table first. We observed that some values are repeated So, to find sum of all we have to multiply weight with number of children and then find the sum.

Let Weight be x and number of children be f

Average =

= = 31.53 kg

∴ Average weight of each child is 31.53 kg

To find the mean let us prepare frequency distribution table first .

Calculate the mid-point of each interval

And put it as x

Mid-point =

Average =

= = = 42.75

To find the mean let us prepare frequency distribution table first.

Calculate the mid-point of each interval

And put it as x

Mid-point =

Average =

= = = 28.625

1^st arrange data in ascending order

6, 9, 11, 14, 15, 18, 20, 22, 25

As we can count there are odd number of terms

∴ Median = term

Where N is number of terms

Median = = = 5^th term = 15

1^st arrange data in ascending order

10, 18, 22, 28, 33, 34, 41, 44, 49, 57, 59, 66

As we count there are odd numbers of terms

∴ Median = Average of term and term

Where N is number of terms which is 12

∴ Median = Average of term and term

= Average of 6^thterm and 7^th term

= Average of 34 and 41

Average =

Sum of terms = 34 + 41 = 75

Average = = 37.5

The given distribution is in inclusive form. It should be converted in exclusive form

Upper limit of 1^st interval is 119

Lower limit of 2^nd interval is 120

= = = 0.5

Actual upper limit = Stated upper limit +

Actual lower limit = Stated lower limit –

As we have sum of frequency to be (N) 50

As it is an even number

It has 2 middle scores

=

For finding 25^th and 26^th term we need to find cumulative frequency

25 and 26 can be covered under Cumulative frequency 29

∴ 129.5 – 139.5 is Median class

⇒ Low real limit (LRL) = 129.5

⇒ Frequency of median class (f_m) = 15

⇒ Cumulative Frequency of above median class (f_c) = 14

⇒ Size of class interval (i) = 10

Median = LRL +

= 129.5 +

= 129.5 +

= 129.5 + = 129.5 + 7.33 = 136.83

As we have sum of frequency to be (N) 40

As it is an even number

It has 2 middle scores

=

For finding 20^th and 21^th term we need to find cumulative frequency

20 and 21 can be covered under Cumulative frequency 27

∴ 15-20 is Median class

⇒ Low real limit (LRL) = 15

⇒ Frequency of median class (f_m) = 10

⇒ Cumulative Frequency of above median class (f_c) = 17

⇒ Size of class interval (i) = 5

Median = LRL +

= 15 +

= 15 +

= 15 + = 15 + 1.66 = 16.66

In the given data

Only 3 is repeater is twice

∴ 3 is mode of given data

In the given data

22 is repeater thrice and 42 is repeated twice

∴ 22 is mode of given data

In the given data

The maximum frequency is 16

Which is of number 20

Hence;

Number 20 is repeated maximum times

∴ 20 is the mode of the data

The width of this interval is 5, i.e., 0,1,2,3,4,5.

We know that,

⇒ midpoint = 14.5

By Definition,

In a set of data, the range is the difference between the highest and the lowest observation.

By definition,

The number of times a particular observation (score) occurs in a data is called its frequency.

In inclusive form,

Actual lower limit = lower limit -0.5

= 0-0.5

= -0.5

And, Actual upper limit = upper limit-0.5

= 4-0.5

= 3.5

During Representation,

The height of a rectangle in a histogram represents the frequency.

During the representation,

In a histogram, the width of the rectangle indicates the class interval.

⇒ Mean = 13.4

Class interval grouping of data is done when the range of data is large.

⇒ 40 = x+27

⇒ x = 13

We know that,

⇒ = 120

The first three multiples of 5 are-5, 10, 15.

And, their mean will be-

⇒ Mean = 5

Arranging the data in ascending order, we get-

29, 32, 37, 40, 42, 70, 83

And, since the no. of observations(n) is 7 and which is odd

⇒ Median = 4^th term

⇒ Median = 40

In inclusive form,

Lower real limit = lower Limit-0.5

= 10-0.5

= 9.5

In inclusive form,

Lower real limit = lower Limit

= 10

Mode of observations is the data with highest frequency.

Here, 3 appears 3 times -

⇒ mode = 3

Mode of observations is the data with highest frequency and here 18 has highest frequency i.e., 20.

⇒ Mode = 18

A collection of data having more than 3 modes is said to be multimode.

(i) The frequency corresponding to the class interval 20-30 indicate that there are 10 values lying between 20 and 30 and which is highest of all.

(ii) 10 will be included in 10-20, 20 will be included in 20-30 and 30 will be included in 30-40.

(iii) Since the intervals includes data values from 10 to 59.

⇒ Range = 59-10

⇒ Range = 49

(i) Mid-points of 0-4, 5-9, 10-14, 15-19, 20-24 are 2, 7, 12, 17, 22 respectively.

(ii) Here maximum frequency is 12 which is corresponding to the 15-19 class interval.

(iii) Range = Highest data value-lowest data value

⇒ Range = 24-3 = 21

(i)

⇒ Mean = 73.17

(ii)

Since, each student got 4 grace marks implies we need to add 4 marks 12 times.

⇒Mean = 77.17

⇒ 100 = x+83

⇒ x = 17

⇒ Mean = 14.75