CLASS X : CHAPTER - 14 STATISTICS

CLASS X : CHAPTER - 14 STATISTICS

IMPORTANT FORMULAS & CONCEPTS

Measures of Central Tendency
In many real-life situations, it is helpful to describe data by a single number that is most representative of the entire collection of numbers. Such a number is called a measure of central tendency. The most commonly used measures are:

1. Mean: The average of ‘n’ numbers is the sum of the numbers divided by n.
2. Median: The middle number when the numbers are written in order. If n is even, the median is the average of the two middle numbers.
3. Mode: The number that occurs most frequently. If two numbers tie for most frequent occurrence, the collection has two modes and is called bimodal.

MEAN OF GROUPED DATA
1. Direct Method:
\[ \text{Mean}, \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] 2. Assumed Mean Method (Short-cut Method):
\[ \text{Mean}, \bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \] where \( d_i = x_i - A \)
3. Step Deviation Method:
\[ \text{Mean}, \bar{x} = A + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h \] where \( u_i = \frac{x_i - A}{h} \)

MODE OF GROUPED DATA
\[ \text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \] where:

• \( l \) = lower limit of the modal class
• \( h \) = size of the class interval (assuming all class sizes to be equal)
• \( f_1 \) = 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

MEDIAN OF GROUPED DATA
\[ \text{Median} = l + \left( \frac{\frac{n}{2} - cf}{f} \right) \times h \] where:

• \( l \) = lower limit of median class
• \( n \) = number of observations
• \( cf \) = cumulative frequency of class preceding the median class
• \( f \) = frequency of median class
• \( h \) = class size (assuming class size to be equal)

Empirical Formula
3 Median = Mode + 2 Mean

Cumulative Frequency Curve (Ogive)
There are three methods of drawing ogive:

• Less Than Method: Convert the series into a 'less than' cumulative frequency distribution. Plot points using upper limits and corresponding cumulative frequencies.
• More Than Method: Convert the series into a 'more than' cumulative frequency distribution. Plot points using lower limits and corresponding cumulative frequencies.
• Less Than and More Than Ogive Method: The median of grouped data can be obtained graphically as the x-coordinate of the point of intersection of the two ogives for this data.,,

MCQ WORKSHEET-I

1. For a frequency distribution, mean, median and mode are connected by the relation
   1. mode = 3mean – 2median
   2. mode = 2median – 3mean
   3. mode = 3median – 2mean
   4. mode = 3median + 2mean
2. Which measure of central tendency is given by the x – coordinate of the point of intersection of the more than ogive and less than ogive?
   1. mode
   2. median
   3. mean
   4. all the above three measures
3. The class mark of a class interval is
   1. upper limit + lower limit
   2. upper limit – lower limit
   3. 1/2 (upper limit + lower limit)
   4. 1/2 (upper limit – lower limit)
4. Construction of cumulative frequency table is useful in determining the
   1. mode
   2. median
   3. mean
   4. all the above three measures
5. For the following distribution, the modal class is:

   Marks (Below)	Number of Students
   10	3
   20	12
   30	27
   40	57
   50	75
   60	80

   1. 10 – 20
   2. 20 – 30
   3. 30 – 40
   4. 40 – 50
6. For the following distribution, the median class is:

   Marks (Below)	Number of Students
   10	3
   20	12
   30	27
   40	57
   50	75
   60	80

   1. 10 – 20
   2. 20 – 30
   3. 30 – 40
   4. 40 – 50
7. In a continuous frequency distribution, the median of the data is 24. If each item is increased by 2, then the new median will be
   1. 24
   2. 26
   3. 12
   4. 48
8. In a grouped frequency distribution, the mid values of the classes are used to measure which of the following central tendency?
   1. mode
   2. median
   3. mean
   4. all the above three measures
9. Which of the following is not a measure of central tendency of a statistical data?
   1. mode
   2. median
   3. mean
   4. range
10. Weights of 40 eggs were recorded as given below. The lower limit of the median class is:

    Weights (in gms)	85 – 89	90 – 94	95 – 99	100 – 104	105 – 109
    No. of Eggs	10	12	12	4	2

    1. 90
    2. 95
    3. 94.5
    4. 89.5

MCQ WORKSHEET-II

1. The median class of the following distribution is:

   C.I.	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60
   F	8	10	12	22	30	18

   1. 10 – 20
   2. 20 – 30
   3. 30 – 40
   4. 40 – 50
2. Weights of 40 eggs were recorded as given below. The lower limit of the modal class is:

   Weights (in gms)	85 – 89	90 – 94	95 – 99	100 – 104	105 – 109
   No. of Eggs	10	12	12	4	2

   1. 90
   2. 95
   3. 94.5
   4. 89.5
3. The arithmetic mean of 12 observations is 7.5. If the arithmetic mean of 7 of these observations is 6.5, the mean of the remaining observations is
   1. 5.5
   2. 8.5
   3. 8.9
   4. 9.2
4. In a continuous frequency distribution, the mean of the data is 25. If each item is increased by 5, then the new median will be
   1. 25
   2. 30
   3. 20
   4. none of these
5. In a continuous frequency distribution with usual notations, if \( l = 32.5, f_1 = 15, f_0 = 12, f_2 = 8 \) and \( h = 8 \), then the mode of the data is
   1. 32.5
   2. 33.5
   3. 33.9
   4. 34.9
6. The arithmetic mean of the following frequency distribution is 25, then the value of p is:

   C.I.	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
   F	5	8	15	p	6

   1. 12
   2. 16
   3. 18
   4. 20
7. If the mean of the following frequency distribution is 54, then the value of p is:

   C.I.	0 – 20	20 – 40	40 – 60	60 – 80	80 – 100
   F	7	p	10	9	13

1. 12
2. 16
3. 18
4. 11

The mean of the following frequency distribution is:

C.I.	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
F	12	16	6	7	9

1. 12
2. 16
3. 22
4. 20

The mean of the following frequency distribution is:

C.I	0–10	10 – 20	20 – 30	30 – 40	40 – 50
F	7	8	12	13	10

1. 12.2
2. 16.2
3. 22.2
4. 27.2

The median of the following frequency distribution is:

C.I	100–150	150 – 200	200 – 250	250 – 300	300 – 350
F	6	3	5	20	10

1. 120
2. 160
3. 220
4. 270

MCQ WORKSHEET-III

1. The range of the data 14, 27, 29, 61, 45, 15, 9, 18 is
   1. 61
   2. 52
   3. 47
   4. 53
2. The class mark of the class 120 – 150 is
   1. 120
   2. 130
   3. 135
   4. 150
3. The class mark of a class is 10 and its class width is 6. The lower limit of the class is
   1. 5
   2. 7
   3. 8
   4. 10
4. In a frequency distribution, the class width is 4 and the lower limit of first class is 10. If there are six classes, the upper limit of last class is
   1. 22
   2. 26
   3. 30
   4. 34
5. The class marks of a distribution are 15, 20, 25,…….45. The class corresponding to 45 is
   1. 12.5 – 17.5
   2. 22.5 – 27.5
   3. 42.5 – 47.5
   4. none of these
6. The number of students in which two classes are equal (referring to a bar graph not fully described in text):
   
   1. VI and VIII
   2. VI and VII
   3. VII and VIII
   4. none of these
7. The mean of first five prime numbers is
   1. 5.0
   2. 4.5
   3. 5.6
   4. 6.5
8. The mean of first ten multiples of 7 is
   1. 35.0
   2. 36.5
   3. 38.5
   4. 39.2
9. The mean of \( x + 3, x – 2, x + 5, x + 7 \) and \( x + 72 \) is
   1. \( x + 5 \)
   2. \( x + 2 \)
   3. \( x + 3 \)
   4. \( x + 7 \)
10. If the mean of n observations \( x_1, x_2, x_3, \dots x_n \) is \( \bar{x} \) then \( \sum_{i=1}^{n} (x_i - \bar{x}) \) is
    1. 1
    2. –1
    3. 0
    4. cannot be found
11. The mean of 10 observations is 42. If each observation in the data is decreased by 12, the new mean of the data is
    1. 12
    2. 15
    3. 30
    4. 54
12. The median of 10, 12, 14, 16, 18, 20 is
    1. 12
    2. 14
    3. 15
    4. 16
13. If the median of 12, 13, 16, x + 2, x + 4, 28, 30, 32 is 23, when x + 2, x + 4 lie between 16 and 30, then the value of x is
    1. 18
    2. 19
    3. 20
    4. 22
14. If the mode of 12, 16, 19, 16, x, 12, 16, 19, 12 is 16, then the value of x is
    1. 12
    2. 16
    3. 19
    4. 18
15. The mean of the following data is
    

    x	5	10	15	20	25
    F	3	5	8	3	1

    1. 12
    2. 13
    3. 13.5
    4. 13.6
16. The mean of 10 numbers is 15 and that of another 20 numbers is 24 then the mean of all 30 observations is
    1. 20
    2. 15
    3. 21
    4. 24

MCQ WORKSHEET-IV

1. Construction of cumulative frequency table is useful in determining the
   1. mean
   2. median
   3. mode
   4. all three
2. In the formula \( \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i} \), finding the mean of the grouped data, \( d_i \)’s are deviations from assumed mean ‘a’ of
   1. lower limits of classes
   2. upper limits of classes
   3. class marks
   4. frequencies of the classes
3. If \( x_i \)’s are the midpoints of the class intervals of grouped data, \( f_i \)’s are the corresponding frequencies and \( \bar{x} \) is the mean, then \( \sum f_i (x_i - \bar{x}) \) is equal to
   1. 0
   2. –1
   3. 1
   4. 2
4. In the formula \( \bar{x} = a + \left( \frac{\sum f_i u_i}{\sum f_i} \right) \times h \), finding the mean of the grouped data, \( u_i = \)
   1. \( (x_i + a)/h \)
   2. \( (x_i - a)/h \)
   3. \( (a - x_i)/h \)
   4. \( (h - x_i)/a \)
5. For the following distribution
   

   Class	0-5	5-10	10-15	15-20	20-25
   Frequency	10	15	12	20	9

the sum of lower limits of the median class and the modal class is

1. 15
2. 25
3. 30
4. 35

Consider the following frequency distribution

Class	0-9	10-19	20-29	30-39	40-49
Frequency	13	10	15	8	11