After studying the chapter Statistics[Class 9] you will be able to calculate range and coefficient of range for given data; to find out quartile deviation for regrouped and grouped data; to calculate mean deviation for regrouped data and grouped data; to draw histogram of varying width for inclusive and exclusive class intervals; to draw commutative frequency curve and identity quartile and median n it; to construct frequency polygons for inclusive and exclusive class intervals; to identify random experiment and types of probability.

## 1.5.1 Introduction – Statistics[Class 9]

The collection of numerical facts with particular information during experiment is called **data.** These data can be grouped into a table that displays frequencies of scores corresponding to various class interval. While grouping the data, if the end points o the groups do not overlap we call it inclusive method if grouping the data and if the end points of consecutive groups overlap, we calling it exclusive method.

**Range – Statistics[Class 9]:**

The difference between the highest and the lowest scores in a given distribution is called range.

### Mean – Statistics**[Class 9]**:

It is the average of the scores, which s equal to the one of the scores divided by the number of scores.

For ungrouped data, the mean is calculated using the formula,

Ẍ = ^{⅀}^{x}/_{N}

For grouped data, the mean is given by Ẍ = ^{⅀}^{fx}/_{N}

### Median – Statistics[Class 9]:

Median is the middle most score in a given set of scores. In ungrouped data, median is the middle score( when the scores are odd) or the average of two middle scores (when the scores are even), after the scores being arranged in ascending or descending order.

Median for grouped data is calculated using the formula,

### Mode – Statistics[Class 9]:

Mode is the score that occurs frequently in a given set of scores. Most repeated score in a ungrouped data is the mode. Mode of the value around which the other scores cluster around densely. In a grouped data, the scores corresponding to the maximum frequency is the mode.

A collection of data can have more than one mode. If the data has only one mode, we say it has Uni mode, if it has 2 modes, we say it has bi mode and it has more than 3 modes, we say it has multi-mode.

## 1.5.2 Measures of dispersion – Statistics[Class 9]:

There are four measures of dispersion viz.

- range(R)
- Quartile Deviation(QD)
- Mean Deviation(MD)
- Standard deviation(SD)

### (a) Range – Statistics[Class 9]:

To understand the range, consider the following set of data:

24, 52, 35, 28, 49, 21

Find out the highest and the lowest scores. Have you observed the highest scores is 52 and the lowest is 21? Take the differenece of these two scores. It is 52 – 21 = 31. What is the difference called? This difference os called **range**.

Example : Calculate the range from the following data:

Marks | 26 | 38 | 54 | 65 | 72 | 88 |

No. of students | 5 | 10 | 15 | 20 | 25 | 30 |

Solution:

We observe that, highest scores H = 88 and lowest scores L = 26

Therefore, range H – L = 88 – 26 = 62 marks

Range is the simplest measure of dispersion. The difference between the highest and the lowest scores of distribution is called **range**.

**Range = Highest Score (H) – Lowest Score(L)**

### (b) Coefficient of range[Class 9]:

Consider the following example:

The wages of six workers of a factory in rupees are:

1600, 1500, 1750, 1800, 1250, 1400

What is the highest and the lowest wages? The highest is 1800 and the lowest is 1250. Let us calculate the ratio of the difference of the highest and the lowest wages to its sum. It is

^{H }^{–}^{ L}/_{H + L} = ^{1800 }^{–}^{ 1250}/_{1800 + 1250} = ^{550}/_{3050} = 0.18 (approximately)

**Coefficient of range **is a relative measure of dispersion and it is based on the value of range. It is also called the range coefficient of dispersion.

**Coefficient of range **is given by =** ^{H }^{–}^{ L}/_{H + L} **

Example : Calculate the coefficient of range for the following data:

No. of wards | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

No. of houses | 32 | 57 | 28 | 96 | 138 | 90 | 66 | 58 |

Solution:

Here, H = 8 ; L = 1

Hence, coefficient of range = ^{H }^{–}^{ L}/_{H + L} = ^{8-1}/_{8+1} = ^{7}/_{8}

### Merits and demerits of range[Class 9]:

**Merits:**

- It is the simplest measure of dispersion and easy to calculate.
- It does not require special knowledge to understand.
- Its calculation takes less time.

**Demerits:**

- It does not take into account all the scores/items of distribution.
- It is affected by extreme scores.
- It does not indicate the direction of variability.

### (c) Quartile Deviation – Statistics[Class 9]:

The points that divide the distribution in to four equal parts are called **quartiles**. If we take the difference between the third quartile and the first quartile, it gives us a value called inter quartile range. It is equal to

Q_{3} – Q_{1}. Half of this is the Semi-Inter Quartile Range *or *quartile deviation which is (Q_{3} – Q_{1})/_{2} . Quartile deviation is also called the semi-inter quartile range.

**(d) Quartile Deviation for ungrouped data:**

Example: The runs scored by a batsman in five innings are 28, 60, 85, 58, 74, 20, 90. Find Q_{1}, Q_{2}, Q_{3} and quartile deviation.

Solution:

Arranging the scores in ascending we get, 28, 60, 85, 58, 74, 20, 90.

There are 7 scores and n = 7

- First quartile(Q
_{1}) =^{n+1}/_{4}th score =^{7+1}/_{4}= 2^{nd}score = 28 - Median(Q
_{2}) =^{n+1}/_{2}th score =^{7+1}/_{2}= 4^{th}score = 60 - Third quartile(Q
_{3}) =^{3(n+1)}/_{4}=^{3(7+1)}/_{4}th score = 6^{th}score = 85 - Quartile deviation =
^{Q3 }^{–}^{ Q1}/_{2}=^{85 }^{–}^{ 28}/_{2}= 28.5 ≅ 29

**(e) Quartile deviation for grouped data – Statistics[Class 9]: **

**Example: Find the median and quartile deviation for the following data:**

x | 3 | 4 | 5 | 6 | 7 |

f | 12 | 35 | 52 | 41 | 18 |

Solution:

Let us find the commutative frequency for the data given:

x | f | Commutative frequency | Remarks |

3 | 1 | 12 | 12, the first 12 scores correspond to x = 3 |

4 | 35 | 47 | (12 + 35) from 13^{th} to 47^{th} scores correspond to x = 5 |

5 | 52 | 99 | (47+52) from 48^{th} to 99^{th} scores correspond to x = 5 |

6 | 41 | 140 | (99+41) from 100^{th} to 140^{th} scores correspond to x = 6 |

7 | 18 | 158 | (140+18) from 141th to 158^{th} scores correspond to x = 7 |

Observe that, n = 158, the last commutative frequency.

Median = ^{n+1}/_{2} th score = 79.5^{th} score ≅ 80^{th} score

From the column f_{c}, from 48^{th} to 99^{th} score corresponds to x = 5. Hence 80^{th} position = 5. Therefore, median = 5

**To find the Quartile Deviation:**

Q_{1} = ^{n}/_{4} th score = ^{158}/_{4} th score = 39.5 i.e., 40^{th} score

From column f_{c}, 13^{th} to 47^{th} scores correspond to x = 4. Hence the 40^{th} position is 4. Therefore,

Q_{1} = 4

Similarly, ^{3n}/_{4} = ^{3×158}/_{4} = 118.5

Hence, Q_{3} is 119^{th} score. The column for f_{c} shows that 100^{th} to 140^{th} scores correspond to x = 6. Therefore, Q_{3} = 6.

Quartile Deviation = ^{Q3 }^{–}^{ Q1}/_{2} = ^{6 }^{–}^{ 4}/_{2} = 1 mark.

### (e) Quartile deviation for grouped data with class intervals:

Example: The heights of 100 students in 9^{th} standard are given below:

Height(cm) | 100 – 110 | 110 – 120 | 120 – 130 | 130 – 140 | 140 – 150 | 150 – 160 |

No. of Students(f) | 10 | 12 | 16 | 30 | 12 | 20 |

Find quartile deviation.

Solution:

**Step 1: **First let us find the commutative frequency corresponding to the frequencies given.

Height(cm) | f_{1} | f_{c} |

100 – 110 | 10 | 10 |

110 – 120 | 12 | 22 |

120 – 130 | 16 | 38(Q_{1} Class) |

130 – 140 | 30 | 68 |

140 – 150 | 12 | 80 |

150 – 160 | 20 | 100(Q_{3} Class) |

N = 100 |

**Step 2:** To find Q_{1} : Recall the formula to find the median

where LRL = lower classs limit, f_{c} = commutative frequency just above the median class f_{m} = frequency corresponding the median class and I = size of the class interval. Now, to find Q_{1} replace ^{N}/_{2} by ^{N}/_{4} in the formula for median. Thus we get,

Now, find out, ^{N}/_{4} :^{ N}/_{4} = ^{100}/_{4} = 25

Locate 25 in the commutative frequency column. This corresponds to the CI 120 – 130. This is Q_{1} class.

From this class, LRL = 120, f_{c} = 22 , f_{m} = 16 and I = 10. Substituting these values in Q_{1} , we get,

= 120 + (0.1875 x 10)

= 120 + 1.875

= 121.88

Thus, Q_{1} ≅ 122

**Step 3:** To find Q_{3}:

In the median formula, replace, ^{N}/_{2} by ^{3N}/_{4} and follow the same steps as in step 2.

Here, ^{3N}/_{4} = ^{3×100}/_{4} = 75. In the commutative frequency column 75 corresponds to class interval 140 – 150. Therefore,

LRL = 140, f_{c} = 68 , f_{m} = 12 and I = 10

= 140 + (0.58 x 10)

= 120 + 5.8

= 145.8

Thus, Q_{3} ≅ 146

**Step 4:** Now, we can find quartile deviation using the formula:

Quartile deviation = ^{Q3 – Q1}/_{2} = ^{146 – 122}/_{2} = 12.

## Statistics Exercise 1.5.3

**Calculate the range and coefficient of range from the following data.**

**a) The heights of 10 children in cm: 122, 144, 154, 101, 168, 118, 155, 133, 160, 140 **

Solution:

Heights of 10 children in cm: 122, 144, 154, 101, 168, 118, 155,133, 160,140.

Range: H – L = 168 – 101 = 67

Coefficient of Range: ^{H – L}/_{H + L}

= ^{67}/_{168 + 101} = ^{67}/_{269} **= 0.249**

**b) Marks scored by 12 students in a test: 31, 18, 27, 19, 25, 28, 49, 14, 41, 22, 33, 13.**

Solution:

** **Marks scored by 12 students

31, 18, 27, 19, 25, 28, 49, 14, 41, 22, 33, 13.

H = 49; L = 13

Range: H – L = 49 – 13 = 36

Coefficient of Range: ^{H – L}/_{H + L} = ^{36}/_{49 + 13} = ^{36}/_{62} = **0.58. **

**c) ****Number of trees planted in 6 months: 186, 234, 465, 361, 290, 142. **

Solution:

No .of trees planted in 6 months:

186, 234, 465, 361, 290, 142.

H = 465; L = 142

Range: H – L = 465 – 142 = 323

Coefficient of Range: ^{H – L}/_{H + L} = ^{323}/_{465+142} = ^{323}/_{607} = **0.532 **

**State quartile deviation for the following data:**

**a) ****30, 18, 23, 15, 11, 29, 37, 42, 10, 21.**

Solution:

** **Arrange the scores in ascending order:

n=10,

10, 11, 15, 18, 21, 23, 29, 30, 37, 42.

(a) First Q1 = ^{n+1}/_{4}

= ^{10+1}/_{4} = ^{11}/_{4} = 2.75 = 3rd score = **15 **

(ii) Third Quartile

Q3 = ^{3(n+1)}/_{4} = ^{3×11}/_{4} = ^{33}/_{4} = 8.25 = 9th = **37 **

(iii) Quartile Deviation:

= (Q3−Q1)/_{2} = ^{37−15}/_{2} = ^{22}/_{2} = **11**

**b) 3, 5, 8, 10, 12, 7, 5.**

Solution:

3,5,5,7,8,10,12

n = 7

(i) Quartile Q1 = ^{n+1}/_{4} = ^{7+1}/_{4} = ^{8}/_{4} = 2nd score

Q1 = **5 **

(ii) Quartile Q3 = 3^{(}^{n + 1)}/_{4} = 3[(8)/_{4}] = 6th score

Q3 = **10 **

(iii) Quartile Deviation:

= [Q3 − Q1]/_{2} = ^{10 – 5}/_{2} = ^{5}/_{2} = **2.5**

**(c)**

Age | 3 | 6 | 9 | 12 | 15 |

No. of children | 4 | 8 | 11 | 7 | 12 |

Solution:

x | f | commutative frequency |

3 | 4 | 4 |

6 | 8 | 12 |

9 | 11 | 23 |

12 | 7 | 30 |

15 | 12 | 42 |

\ n = 42

Q1 = ^{n}/_{4} ; score = ^{42}/_{4} = 10.5;11th score

From fc ∴Q1 = 6

Q3 = ^{3n}/_{4} = ^{3 ×42}/_{4} = 31.5; 32nd score ∴Q3 = 15

Q.D = [Q3−Q1]/_{2} = ^{15−6}/_{2} = ^{9}/_{2} = **4.5**

**d)**

Marks scored | 10 | 20 | 30 | 40 | 50 | 60 |

No. of students | 12 | 7 | 16 | 08 | 18 | 22 |

Solution:

x | f | fc |

10 | 12 | 12 |

20 | 07 | 19 |

30 | 16 | 35 |

40 | 08 | 43 |

50 | 18 | 61 |

60 | 22 | 83 |

n = 83

Q1= ^{n}/_{4} = ^{83}/_{4} = **20.75; **21st score

∴Q1 = 30

Q3 = ^{3n}/_{4} = ^{3 ×83}/_{4} = 3X20.75 = 62.25; 62nd score

∴Q3 = 60

Q.D = [Q3−Q1]/2 = ^{60 – 30}/_{2} = ^{30}/_{2} = **15 **

∴QD = 15

**Compute quartile deviation for each of the following tables.**

**a)**

Class interval | Frequency | fc |

5 – 15 | 11 | 11 |

15 – 25 | 5 | 16 |

25 – 35 | 15 | 31 |

35 – 45 | 9 | 40 |

45 – 55 | 22 | 62 |

55 – 65 | 8 | 70 |

65 – 75 | 17 | 87 |

Solution:

n = 87

Q1= ^{n}/_{4} = ^{87}/_{4} = **21.75 **

22nd score CI = 25 – 35

∴LRC = 25

fc = 31; i = 10

Q2 = LRL +( [^{N}/_{4}−fc]/_{fm})i

= 25 + [(^{87}/_{4}−16)/_{15}] 10

= 25 + [^{21.75 −16}/_{15}] ×10

**Q****2 ****= 28.83**

Q3 = LRL + [(^{3N}/_{4} – fc)/_{fm}]*i

^{3N}/_{4} = ^{3 × 87}/_{4} = 65.25 class interval 55 – 65

L = 55, fc = 62, fm = 8, CI = 10

LRL = 55 +(^{65.25 – 62})/_{8} ×10

= 55 + 4.06

**LRL = 59.06 **

Quartile Deviation = [Q3 − Q1]/_{2}

= ^{59.06 – 28.83}/_{2}

= ^{30.23}/_{2}

**Q.D = 15.11**

**(b)**

class interval | frequency | fc |

1 – 9 | 4 | 4 |

10 – 19 | 3 | 7 |

20 – 29 | 20 | 27 |

30 – 39 | 12 | 39 |

40 – 49 | 5 | 44 |

50 – 59 | 8 | 52 |

60 – 69 | 14 | 66 |

70 – 79 | 27 | 93 |

80 – 89 | 2 | 95 |

90 – 99 | 5 | 100 |

H = 100

Solution:

n = 100

^{100}/_{4} = 25th Score 20 – 29; LRL = 19.5

Fc = 7; fm = 20

Q1 = LRL+[ (^{N}/_{4}−fc)/_{fm}]*i

= 19.5 + [^{25 – 7}/_{20}]× 10

= 19.5 + ^{18}/_{20}*10

**Q****1 ****= 28.5 **

^{3N}/_{4} = ^{3 }^{× }^{100}/_{4} = 3 × 25 = 75th score cl 70 – 79

LRL = 69.5; fc = 66; fm = 14

Q3 = 69.5 + [^{(}^{75 – 66)}/_{14}] × 10

= 69.5 + 3.33

**Q****3 ****= 72.83**

Quartile Deviation = (Q3 − Q1)/_{2}

= ^{72.83 – 28.5} /_{2}

= ^{44.33}/_{2}

**Q.D = 22.16**

## 1.5.3 Mean Deviation – Statistics[Class 9]:

### Calculation of mean deviation for ungrouped data about median[Class 9]:

Consider the following set of scores:

15, 11, 13,20, 26, 18, 21

arranging scores in ascending order:

11, 13, 15, 18, 20, 21, 26.

Here we have 7 scores which is odd.

Therefore, median = (^{N+1}/_{2}) th score = (^{7+1}/_{2})th score = (^{8}/_{2}) th score = 4^{th} score = 18

Now let us find the deviation of each score from the median. the deviation D = score(X) – median.

Let us take only positive value of D. The positive value of D is called absolute value and denoted by |D|.

Scores(X) | Deviations from median D = X – median | |D| |

11 | 11 – 18 = -7 | 7 |

13 | 13 – 18 = -5 | 5 |

15 | 15 – 18 = -2 | 2 |

18 | 18 – 18 = 0 | 0 |

20 | 20 – 18 = 2 | 2 |

21 | 21 – 18 = 3 | 3 |

26 | 26 – 18 = 8 | 8 |

N = 7 | ⅀|D| = 28 |

Add all |D| and divide it by the total number of scores. It gives you mean deviation.

i.e.,

^{⅀}^{|D|}/_{N} = ^{28}/_{7} = 4

Therefore, mean deviation = 4

__Calculation f mean deviation for ungrouped data about mean:__

Example 10: Calculate the mean deviation from the mean for the scores given below:

15, 11, 13, 20, 26, 18, 21

Solution:

Arrange the scores in an order:

11, 13, 15, 18, 20, 21, 26

We known mean = ^{sum of all the scores}/_{number of scores} = ^{⅀}^{x}/_{N}

= ^{11+13+15+18+20+21+26}/_{7} = ^{124}/_{7} ≅17.7

Now calculate the deviation D of each score from mean and find out ⅀|D|.

Scores(X) | Deviations from mean D = X – mean | |D| |

11 | 11 -17.7 = -6.7 | 6.7 |

13 | 13 -17.7 = -4.7 | 4.7 |

15 | 15 -17.7 = -2.7 | 2.7 |

18 | 18 -17.7 = 0.3 | 0.3 |

20 | 20 -17.7 = 2.3 | 2.3 |

21 | 21 -17.7 = 3.3 | 3.3 |

26 | 26 -17.7 = 8.3 | 8.3 |

N = 7 | ⅀|D| = 28.3 |

Now mean deviation from mean is = ^{⅀}^{|D|}/_{N} = ^{28.3}/_{7} = 4.04

__Calculation of mean deviation for grouped data about median:__

Example : Calculate the mean deviation for the following data about median.

Class interval | 0 – 4 | 5 – 9 | 10 – 14 | 15 – 19 | 20 – 24 | 25 – 29 |

Frequency | 11 | 12 | 17 | 12 | 20 | 28 |

Solution:

First let us find the median

CI | f | fc | midpoint of CI(X) | Deviation D = X – median | |D| | fx|D| |

0 – 4 | 11 | 11 | 2 | 2 – 18.7 = -16.7 | 16.7 | 183.7 |

5 – 9 | 12 | 23 | 7 | 7 – 18.7 = -11.7 | 11.7 | 140.4 |

10 – 14 | 17 | 40 | 12 | 12 – 18.7 = -6.7 | 6.7 | 113.9 |

15 – 19 | 12 | 5 | 17 | 17 – 18.7 = -1.7 | 1.7 | 20.4 |

20 – 24 | 20 | 72 | 22 | 22 – 18.7 = 3.3 | 3.3 | 66.0 |

25 – 29 | 28 | 100 | 27 | 27 – 18.7 = 8.3 | 8.3 | 232.4 |

N = 100 | ⅀f|D|= 756.8 |

= 14.5 + (^{50 – 40}/_{12}) x 5

≅ 14.5 + (0.83 x 5)

= 14.5 + 4.15

= 18.65

≅ 18.7

After finding the median , the deviation of median from midpoint of the class intervals are calculated. This gives D. Frequency (f) is multiplied with |D| to get f|D|. By adding all F|D|, we get⅀f|D|. these are shown above.

Mean deviation = ^{⅀f}^{|D|}/_{N} = ^{756.8}/_{100} = 7.57

__Calculation of mean deviation for grouped data about mean:__

Example 12: Calclate the mean deviation for the data given below:

Class interval | 0 – 10 | 10 – 20 | 20 – 30 | 30 – 40 | 40 – 50 |

Frequency | 5 | 3 | 9 | 12 | 6 |

Solution:

Class interval | Frequency | midpoint x | fx | Deviation D = x – X | |D| | f|D| |

0 – 10 | 5 | 5 | 25 | 5 – 28.1 = – 23.1 | 23.1 | 115.5 |

10 – 20 | 3 | 15 | 45 | 15 – 28.1 = – 13.1 | 13.1 | 39.3 |

20 – 30 | 9 | 25 | 225 | 25 – 28.1 = – 3.1 | 3.1 | 27.9 |

30 – 40 | 12 | 35 | 420 | 35 – 28.1 = 6.9 | 6.9 | 82.8 |

40 – 50 | 6 | 45 | 270 | 45 – 28.1 = 16.9 | 16.9 | 101.4 |

⅀fx | ⅀f|D| = 366.9 |

Mean x = ^{⅀fx}/_{N} = ^{985}/_{35} = 28.1

Mean deviation = ^{⅀f}^{|D|}/_{N} = ^{366.9}/_{35} = 10.4

## Statistics[Class 9] Exercise 1.5.3

**Find the mean deviation about mean for the following date:**

**a) 14, 21, 28, 21, 18 **

Solution:

Mean = ^{14+21+28+21+18}/_{5} = ^{102}/_{5} = 20.4

Score | Deviation from mean | |D| |

14 | 14 – 10.4 = -6.4 | 6.4 |

18 | 18 – 20.4 = -2.4 | 2.4 |

21 | 21 – 20.4 = 0.6 | 0.6 |

21 | 21 – 20.4 = 0.6 | 0.6 |

28 | 28 – 20.4 = 7.6 | 7.6 |

(b)

Score(x) | 6 | 20 | 8 | 18 | 16 | 12 | 14 | 10 |

Frequency(f) | 2 | 7 | 11 | 27 | 18 | 13 | 17 | 5 |

Solution:

x | f | fxD= x – x | |D| | f|D| |

6 | 2 | 12 – 8.58 | 8.58 | 17.16 |

20 | 7 | 140 + 5.42 | 5.42 | 37.94 |

8 | 11 | 88 – 6.58 | 6.58 | 72.38 |

18 | 27 | 486 – 3.42 | 3.42 | 92.34 |

16 | 18 | 288 – 1.42 | 1.42 | 25.56 |

12 | 13 | 156 – 2.58 | 2.58 | 33.56 |

14 | 17 | 238 – 0.58 | 0.58 | 9.86 |

10 | 5 | 50 – 4.58 | 4.58 | 22.9 |

N = 100 | 1458 | 311.68 |

**Find the mean deviation about mean for the following data:**

**a) 15, 18, 13, 16, 12, 24, 10, 20**

Solution:

15+ 18+ 13+ 16+ 12+ 24+ 10+ 20 =128

N = 8

⅀x = 1258

Mean= ^{⅀}^{x}/_{N} =^{128}/_{8} = 16

x | D = x – x | |D| |

10 | 10-16=-6 | 6 |

12 | 12-16 = -4 | 4 |

13 | 13 – 16 = -3 | 3 |

15 | 15 – 16 = -1 | 1 |

16 | 16 – 16 = 0 | 0 |

18 | 18 – 16 = 2 | 2 |

20 | 20 – 16 = 4 | 4 |

24 | 24 – 16 = 8 | 8 |

⅀28 |

MD =^{⅀}^{|D|}/_{N} = ^{28}/_{8} = 3.5

**(b)**

CI | f |

10-19 | 6 |

20-29 | 4 |

30-39 | 10 |

40-49 | 9 |

50-59 | 11 |

60-69 | 8 |

70-79 | 2 |

Solution:

CI | f | x | fx | D = x – x | |D| |

10-19 | 6 | 14.5 | 87 | -29.5 | 177 |

20-29 | 4 | 24.5 | 98 | -19.5 | 78 |

30-39 | 10 | 34.5 | 345 | -9.5 | 95 |

40-49 | 9 | 44.5 | 400.5 | 0.5 | 4.5 |

50-59 | 11 | 54.5 | 599.5 | 10.5 | 115.5 |

60-69 | 8 | 64.5 | 516 | 20.5 | 164.0 |

70-79 | 2 | 74.5 | 149 | 30.5 | 61 |

N = 50 | ⅀fx=2195 | ⅀|D|=695 |

(c)

Class interval | Frequency |

0-5 | 9 |

5-10 | 13 |

10-15 | 6 |

15-20 | 12 |

20-25 | 9 |

25-30 | 6 |

30-35 | 10 |

35-40 | 15 |

40-45 | 6 |

45-50 | 4 |

Solution:

Class interval | Frequency | x | fx | D= x – x | |D| |

0-5 | 9 | 2.5 | 22.5 | -20.5 | 184.5 |

5-10 | 13 | 7.5 | 97.5 | -15.5 | 201.5 |

10-15 | 6 | 12.5 | 75.0 | -10.5 | 63.0 |

15-20 | 12 | 17.5 | 210.0 | -5.5 | 66.0 |

20-25 | 9 | 22.5 | 202.5 | -0.5 | 4.5 |

25-30 | 6 | 27.5 | 165.0 | -4.5 | 27.0 |

30-35 | 10 | 32.5 | 325.0 | 9.5 | 95.0 |

35-40 | 15 | 37.5 | 562.5 | 14.5 | 217.5 |

40-45 | 6 | 42.5 | 255.0 | 19.5 | 117.0 |

45-50 | 4 | 47.5 | 190.0 | 24.5 | 98.0 |

N = 90 | ⅀fx=2105 | ⅀|D|=1074.0 |

**Find the mean deviation about median for the following data:**

**a) 18, 23, 9, 11, 26, 4, 14, 21 **

Solution:

4, 9, 1, 14, 18, 21, 23, 26

Median = ^{N+1}/_{2} = ^{8+1}/_{2} = ^{9}/_{2} = 4.5^{th}

^{14+18}/_{2} = ^{32}/_{2}= 16

median = 16

x | D = x – median | |D| |

4 | 4 – 16 = -12 | 12 |

9 | 9 – 16 = -7 | 7 |

11 | 11 – 16 = -2 | 5 |

14 | 14 – 16 = -2 | 2 |

18 | 18 – 16 = 2 | 2 |

21 | 21 – 16 = 05 | 5 |

23 | 23 – 16 = 07 | 7 |

26 | 26 – 16 = 10 | 10 |

50 |

MD = ^{50}/_{8} = 6.25

(**b)**

Class interval | Frequency |

8 – 12 | 14 |

13 – 17 | 8 |

18 – 22 | 20 |

23 – 27 | 7 |

28 – 32 | 11 |

33 – 37 | 10 |

38 – 42 | 24 |

43 – 47 | 6 |

Solution:

Class interval | Frequency | x | fx | D = x – median | f|D| |

8 – 12 | 14 | 10 | 14 | 10 – 23 = -13 | 182 |

13 – 17 | 8 | 15 | 22 | 15 – 23 = -8 | 64 |

18 – 22 | 20 | 20 | 42 | 20 – 23 = -3 | 60 |

23 – 27 | 7 | 25 | 29 | 25 – 23 = 2 | 14 |

28 – 32 | 11 | 30 | 60 | 30 – 23 = 7 | 77 |

33 – 37 | 10 | 35 | 70 | 35 – 23 = 12 | 120 |

38 – 42 | 24 | 40 | 94 | 40 – 23 = 17 | 408 |

43 – 47 | 6 | 45 | 100 | 45 – 23 = 22 | 132 |

N = 100 | ⅀f|D|=1057 |

**(c) **

Class interval | frequency |

20 – 30 | 9 |

30 – 40 | 18 |

40 – 50 | 7 |

50 – 60 | 21 |

60 – 70 | 11 |

70 – 80 | 4 |

Solution:

Class interval | frequency | x | fc | D = x – median | f|D| |

20 – 30 | 9 | 25 | 19 | 25-42=-17 | 153 |

30 – 40 | 18 | 35 | 27 | 35-42=-7 | 126 |

40 – 50 | 7 | 45 | 34 | 45-42=3 | 21 |

50 – 60 | 21 | 55 | 55 | 55-42=13 | 273 |

60 – 70 | 11 | 65 | 66 | 65-42 = 23 | 253 |

70 – 80 | 4 | 75 | 70 | 75-42 =33 | 132 |

N = 70 | ⅀f|D|=958 |

**Find the mean deviation about mean and median for the following data:**

**a)**

Cl | 1-5 | 6-10 | 11-15 | 16-20 | 21-25 |

f | 2 | 9 | 5 | 4 | 10 |

Solution:

Cl | f | fc | x | fx | D = x – x | f|D| |

1-5 | 2 | 2 | 3 | 06 | 3 – 15 = -12 | 24 |

6-10 | 9 | 11 | 8 | 72 | 8 – 15 = -7 | 63 |

11-15 | 5 | 16 | 13 | 65 | 13 – 15 = -2 | 10 |

16-20 | 4 | 20 | 18 | 72 | 18-15 = 3 | 12 |

21-25 | 10 | 30 | 23 | 230 | 23 – 15 = 8 | 80 |

N=30 | ⅀fx=445 | 189 |

Mean = ^{445}/_{30} = 14.83 = 15

MD from Mean = ^{189}/_{30} = 6.3

CI | f | fc | x | D | fx |

1-5 | 2 | 2 | 3 | -11.5 | 23 |

6-10 | 9 | 11 | 8 | -6.5 | 58.5 |

11-15 | 5 | 16 | 13 | -1.5 | 7.5 |

16-20 | 4 | 20 | 18 | ±3.5 | 14.0 |

21-25 | 10 | 30 | 23 | ±8.5 | 85.0 |

N=30 | ⅀fx = 188 |

**(b)**

CI | 5 – 10 | 10-15 | 15-20 | 20-25 | 25-30 |

f | 5 | 12 | 3 | 11 | 9 |

Solution:

CI | f | fc | x | fx | D=x-x | f|D| | x-median | f|D| |

5-10 | 5 | 5 | 7.5 | 37.5 | 7.5-18=10.5 | 52.5 | -7.5 | 62.5 |

10-15 | 12 | 17 | 12.5 | 150 | -5.5 | 66 | -2.5 | 90 |

15-20 | 3 | 20 | 17.5 | 52.5 | -0.5 | 1.5 | 2.5 | 7.5 |

20-25 | 11 | 31 | 22.5 | 247.5 | 4.5 | 49.5 | 7.5 | 29.5 |

25-30 | 9 | 40 | 27.5 | 247.5 | 9.5 | 85.5 | 12.5 | 0.5 |

N= 40 | ⅀fx=735 | 235 | 255 |

Mean = ^{735}/_{40} = 18.375 ≈18

## 1.5.4 Graphical Representation – Statistics[Class 9]:

### 1. Construction and Interpretation of Histogram- Statistics[Class 9]:

- Histogram is the most properly and widely used methods of graphical representations.
- Histogram is a two dimensional graphical representation of a continuous frequency distribution.
- In a histogram the area of rectangular are proportional to the frequencies.

Class intervals are marked on the x – axis and frequencies on the Y – axis. Class intervals must be exclusive. IF the class intervals are in inclusive form, they are to be converted into exclusive form. Rectangles of width equal to class interval and length equal to frequencies re drawn.

### (a) Histograms of varying width:

The width of each class interval is calculated by the corresponding frequencies. This is done by using a concept Frequency Density.

Frequency density is the ratio of frequency and its class width. when we consider the frequency density the length of the rectangle is to be modified accordingly. Length of the rectangle is the product of frequency density and the minimum class width of given data.

Length of the rectangle = ^{frequency}/_{class width} X C

Here C is the minimum class width of the given data.

### Commutative Frequency Curve – Statistics[Class 9]:

Commutative frequency curve is a graph drawn with commutative frequency against the upper limit of class interval. The points are joined by a smooth curve and the curve is joined to the lower limit of the first class interval.

### Frequency Polygon – Statistics[Class 9]:

Frequency polygon is also a graphical representation of data, where the frequency is plotted against midpoint of the class interval. The frequencies corresponding to the mid points of class intervals are joined by line segments to get the frequency polygon.

A frequency polygon is drawn by drawing a histogram for the given data and joining the midpoints of the top of the rectangles. It can be drawn by marking the midpoints of the class intervals corresponding to their respective frequencies and joining them by line segments.

## Statistics[Class 9] Exercise 1.5.4

**Construct histogram of variable width for the following data:**

**a)**

CI | 25-29 | 30-35 | 36-40 | 41-50 | 51-56 | 57-60 |

f | 10 | 24 | 15 | 20 | 12 | 16 |

Solution:

CI | 24.5-29 | 29.5-35.5 | 35.5 – 40.5 | 40.5-50.5 | 50.5-56.5 | 56.5-60.5 |

f | 10 | 24 | 15 | 20 | 12 | 16 |

class width | 5 | 6 | 5 | 10 | 6 | 4 |

length of the rectangles | 10 | 20 | 15 | 10 | 10 | 20 |

**(b)**

CI | 0 – 10 | 10- 15 | 15 – 20 | 20 – 30 | 30 – 40 | 40 – 60 | 60-70 |

f | 20 | 15 | 10 | 25 | 5 | 30 | 50 |

Solution:

CI | 0 – 10 | 10- 15 | 15 – 20 | 20 – 30 | 30 – 40 | 40 – 60 | 60-70 |

f | 20 | 15 | 10 | 25 | 5 | 30 | 50 |

class width | 10 | 5 | 5 | 10 | 10 | 20 | 10 |

Length of the rectangles | 10 | 15 | 10 | 12.5 | 2.5 | 7.5 | 25 |

**Draw given (cumulative frequency curve) for the data given below:**

Class interval | frequency |

1000 – 1100 | 52 |

1100 – 1200 | 35 |

1200 – 1300 | 25 |

1300 – 1400 | 14 |

1400 – 1500 | 41 |

1500 – 1600 | 33 |

Solution:

Class interval | frequency | Cumulative frequency |

1000 – 1100 | 52 | 52 |

1100 – 1200 | 35 | 87 |

1200 – 1300 | 25 | 112 |

1300 – 1400 | 14 | 126 |

1400 – 1500 | 41 | 167 |

1500 – 1600 | 33 | 200 |

**(b)**

Class interval | Frequency |

5 – 14 | 4 |

15 – 24 | 8 |

25 – 34 | 12 |

35 – 44 | 14 |

45 – 54 | 6 |

55 – 64 | 4 |

65 – 74 | 18 |

75 – 84 | 24 |

Solution:

Class interval | corrective factor | Frequency | cumulative frequency |

5 – 14 | 4.5 – 14.5 | 4 | 4 |

15 – 24 | 14.5 – 24.5 | 8 | 12 |

25 – 34 | 24.5 – 34.5 | 12 | 24 |

35 – 44 | 34.5 – 44.5 | 14 | 38 |

45 – 54 | 44.5 – 54.5 | 6 | 44 |

55 – 64 | 54.5 – 64.5 | 4 | 48 |

65 – 74 | 64.5 – 74.5 | 18 | 66 |

75 – 84 | 74.5 – 84.5 | 24 | 90 |

**Construct frequency polygon for the following data :**

**a)**

CI | 5 – 10 | 10 – 15 | 15 – 20 | 20 – 25 | 25 – 30 | 30 – 35 | 35 – 40 | 40 – 45 |

f | 2 | 5 | 7 | 6 | 1 | 9 | 14 | 8 |

Solution:

**(b)**

CI | 30-35 | 35-40 | 40-45 | 45-50 | 50-55 |

f | 12 | 20 | 16 | 8 | 14 |

Solution:

## 1.5.5 Random Experiment and the concept of Probability – Statistics[Class 9]:

Based on some assumptions , uncertainity can be measured mathematically by what is called probability. Probability has wide applications in the feild of physical science, commerce, biological sciences, weather forecasting, insurance, economics, sociology, investments and in various such other areas.

### (a) Trial – Statistics[Class 9]:

A trial is an action which results in one or more outcomes.

Consider the following examples:

- rolling an unbiased die.
- Picking up a red card from a deck of playing cards.
- Drawing a marble from a bag of different colored marbles.
### (b) Random Experiment – Statistics[Class 9]:

A random experiment is one which exact outcomes are not possible to predict. For example, in tossing a coin we cannot predict outcomes head or tail.

### (c) Sample Space – Statistics[Class 9]:

Consider the event of throwing an unbiased die. The possible outcomes are 1, 2, 3, 4,5 6. The set of all these possible outcomes is called Sample space.

S = {1, 2, 3,4, 5, 6}

### (d) Empirical probability – Statistics[Class 9]:

It is the probability based on actual experiment leading to the possibility of outcomes.

Consider an experiment of tossing a coin 10 times. Let the frequency of head appearing would be 6 and that tail would be 4. Then the empirical probability of appearing head would be ^{6}/_{10} = 0/6 and that of tall would be ^{4}/_{10} = 0.4 .

The empirical probability of a certain event of an experiment is based on the outcomes of actual experiment. For this reason, empirical probability is also known as experimental probability.

If the number of tosses increases, the empirical probability of a head (or also tail)seems to approach the number ^{1}/_{2} or 0.5. This is actually known as the theoretical probability of getting a head (or a tail)

IF n is the number of trails of an event E, then the empirical probability p€ is given by,

P(E) = ^{Number of outcomes favourable to E}/_{Number of possible outcomes to the experiment.}

_{ }

## Statistics[Class 9] – Exercise 1.5.5

**Two unbiased 6 – faced die are thrown. What is the total number out comes?**

**Solution:**

total number of outcomes = 6 × 6 = 36

**A die has the faces numbered 2, 4, 6, 8, 10 and 12. It is thrown once. What is the probability that an even numbered face shows up?**

Solution:

Probability that even number farm shows up = ^{6}/_{6} = 1

**In a pack of 52 playing cards, a card was selected at random. What is the probability that the card selected was both red and black?**

Solution:

Zero: A card cannot be both red and black.

**Weather forecast made for 30 days in a month was recorded and found that it was correct for 21 days. What is the probability that on a randomly selected day, the forecast is****(i) Correct and (ii) Not correct?**

Solution:

Probability of correct forecast = ^{21}/_{30} = ^{7}/_{10}

Probability of correct forecast= ^{9}/_{30} = ^{3}/_{10}