Statistics
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The document discusses measures of central tendency (mode, median, mean) and measures of dispersion (range, quartiles, interquartile range, variance, standard deviation) for both discrete and grouped data. It provides formulas and examples of calculating these statistics for different datasets including examples with raw data, frequency tables, histograms and ogives. It also discusses how to calculate the statistics from incomplete data by completing tables and calculating sums.
![STATISTICS
1. Measures of Central Tendency
Mode, median and mean
For a sample of discrete data, the mode is the observation, x with the highest frequency, f.
1
2N −F
For grouped data in a cumulative frequency table, the median = L +
C
fm
where L is the lower boundary of the median class
N is total frequency
F is the cumulative frequency before the median class
C is the interval of the median class
fm is the frequency of the median class
∑ x . For grouped data, x =
The mean is the average of all the observations. Hence, x =
N
(a)
Find the mode, median and mean for
2, 3, 1, 2, 6, 8, 9, 3, 2, 3.
(b)
[Mode = 2, Median = 3, Mean = 4]
Find the modal class and calculate the median
(d)
and mean for the data in the table below.
Points
frequency
0-4
2
5-9
5
2
1
4
3
6
2
9 12 13
1 2 1
[Mode = 4, Median = 6, Mean = 7.2]
Find the modal class and calculate the median
and mean for the data in the table below.
Marks
10 - 14 15 - 19
8
3
.
Find the mode, median and mean for the data in
the table below.
Score
Frequency
(c)
∑f x
∑f
f
1 - 15 16 - 30 31 - 45 46 - 60 61 - 75 76 - 90
8
11
25
34
16
6
[Modal class = 10-14, Median = 10.75, Mean = 10.33] [Modal class = 46-60, Median = 48.15, Mean = 46.53]
Statistics
1](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-1-320.jpg)
![(e)
Estimate the mode and calculate the median and
mean for the histogram below.
(f)
[Mode = 72.8, Median = 72.71, Mean = 72.6]
Statistics
Estimate the mode and calculate the median and
mean for the histogram below.
[Mode = 152.5, Median = 150.09, Mean = 149.12]
2](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-2-320.jpg)
![(g)
Find the modal class and find the median and
mean for the ogive below.
(h)
[Modal class = 15-19, Median = 17.6, Mean = 18.625]
Statistics
Find the modal class and find the median and
mean for the ogive below.
[Modal class = 45-49, Median = 49.5, Mean = 50.25]
3](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-3-320.jpg)
![2. Measures of Dispersion:
Range, quartiles, interquartile range, variance and standard deviation.
The range of a sample is the difference between the observations with the highest value and the lowest value.
1
1
4N −F
4N −F
The first quartile, Q1 = LQ1 +
CQ1 and the third quartile, Q3 = LQ3 +
CQ3
f Q1
f Q3
Interquartile range = Q3 – Q1. It should be recognised that the median is Q2.
∑ (x − x )
=
2
The variance, σ
2
N
∑x
=
2
2
− x for discrete data.
N
∑ ( f x − x)
=
∑f
2
For grouped data, σ
2
∑ (x
−x
N
)
2
=
∑ ( f x − x)
∑f
∑x
(a)
(c)
2
−x .
2
N
2
For grouped data, σ =
2
variance = σ
The standard deviation is actually
For discrete data, σ =
∑f x
∑f
=
=
2
−x .
∑f x
∑f
2
2
−x .
Find the range and the interquartile for
5, 1, 2, 3, 4, 6, 3, 8, 2, 5, 9.
(b)
[Range = 8, Interquartile range = 4]
Find the range and the interquartile for
(d)
Score
1 4 5 6 8 9
Frequency 1 3 1 1 2 1
[Range = 8, Interquartile range = 4]
Statistics
Find the range and the interquartile range for
12, 17, 13, 19, 15, 8, 12, 11.
[Range = 11, Interquartile range = 4.5]
Find the range and the interquartile range for
Points
2 4 6 8
No. of person
3 5 2 2
[Range = 6, Interquartile range = 3]
4](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-4-320.jpg)
![(e)
Find the interquartile range for the table below.
Marks
No. of
Students
1 - 20
21 - 40
41 - 60
61- 80
4
9
12
10
(f)
Find the interquartile range for the table below.
Age (year)
1-20 21-40 41-60 61-80
No. of residents
66
99
57
28
81 - 100
5
[Interquartile range = 40.77 marks]
(e)
Calculate the interquartile range for the ogive
below.
(f)
[Interquartile range = 12.5 mm]
Statistics
[Interquartile range = 28.95 years]
Calculate the interquartile range for the ogive
below.
[Interquartile range = 6.5 years]
5](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-5-320.jpg)
![(g)
Calculate the interquartile range for the
histogram below.
(h)
[Interquartile range = 5.295 kg]
Statistics
Calculate the interquartile range for the
histogram below.
[Interquartile range = 18.03 minutes]
6](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-6-320.jpg)
![(a)
Find the mean, variance and the standard
deviation for the data below.
5, 12, 6, 3, 6, 10.
(b)
Find the mean, variance and the standard
deviation for the data below.
18, 12, 16, 11, 19, 18, 12, 14.
[Mean = 15, 92 = 7.5, 9 = 2.739]
Complete the table below and calculate the
mean, variance and the standard deviation for
the data.
x
2
4
6
8
10 12
f
1
2
2
2
1
2
fx
fx
x−x
x−x
(x − x )2
2
f (x − x )
(c)
[Mean = 7, 92 = 9.333, 9 = 3.055]
Complete the table below and calculate the
(d)
mean, variance and the standard deviation for
the data.
x
1
2
3
4
5
6
f
1
3
4
7
3
2
(x − x )2
2
f (x − x )
[Mean = 7, 92 = 9.333, 9 = 3.055]
Statistics
[Mean = 7.2, 92 = 10.66, 9 = 3.265]
7](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-7-320.jpg)
![(e)
Complete the table below and calculate the
mean, variance and the standard deviation for
the data.
Class 1 - 3 4 - 6 7 - 9 10 - 12 13 - 15
f
1
3
8
6
2
(f)
Complete the table below and calculate the
mean, variance and the standard deviation for
the data.
Class 10 - 19 20 - 29 30 - 39 40 - 49
f
5
7
5
3
x
x
fx
fx
x2
x2
fx 2
fx 2
[Mean = 8.75, 92 = 8.8875, 9 = 2.981]
[Mean = 27.5, 92 = 101, 9 = 10.05]
(g) Complete the table below and calculate the
(h) Complete the table below and calculate the
mean, variance and the standard deviation for
mean, variance and the standard deviation for
the data.
the data.
Class 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25
Class 0 - 19 20 - 39 40 - 59 60 - 79
f
3
13
23
9
2
f
4
7
6
3
x
x
fx
fx
2
x2
fx 2
fx 2
x
[Mean = 12.4, 92 = 20.64, 9 = 4.543]
Statistics
[Mean = 37.5, 92 = 376, 9 = 19.39]
8](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-8-320.jpg)
![(i)
(j)
Extract the data from the histogram above and
complete the table below. Calculate the mean,
variance and the standard deviation for the data.
Extract the data from the histogram above and
complete the table below. Calculate the mean,
variance and the standard deviation for the data.
Class
x
f
fx
fx 2
Class
[Mean = 55.55, 92 = 41.6475, 9 = 6.453]
Statistics
x
f
fx
fx 2
[Mean = 150.3, 92 = 172.36, 9 = 13.13]
9](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-9-320.jpg)
![1. SPM 2003 Paper 2 Compulsory Question No. 5
A set of test marks: x1, x2, x3, x4, x5, x6 has a mean value of 5 and standard deviation of 1.5.
(a) Find the
(i) sum of marks, x,
(ii) sum of squared marks, x2.
(b) Each mark is multiplied by 2 and then 3 is added.
For the set of new marks, find the
(i) mean,
(ii) variance.
[3]
[4]
Answer:
(a) (i) 30 (ii) 163.50 (b) (i) 13 (ii) 3
_________________________________________________________________________________________
2. SPM 2004 Paper 2 Compulsory Question No. 4
A set of data contains 10 numbers. The sum of these numbers is 150
and the sum of their squares is 2472.
(a) Find the mean and variance for these 10 numbers.
(b) Another number is added to this set of data and the mean increase by 1.
Find
(i) the value of this number,
(ii) the standard deviation for the set of 11 numbers.
[3]
[4]
Answer:
(a) mean = 15, variance = 22.2 (b) (i) 26 (ii) 5.494
_________________________________________________________________________________________
3. SPM 2005 Paper 1 Compulsory Question No. 23
The mean of four numbers is m .
The sum of the squares of the numbers is 100 and the standard deviation is 3k.
Express m in terms of k.
[3]
Answer:
25 - 9k2
_________________________________________________________________________________________
4. SPM 2005 Paper 2 Compulsory Question No. 4
Number of Pupils
The diagram beside is a histogram which represents the
distribution of the marks obtained by 40 pupils in a test.
(a) Without using an ogive,
calculate the median mark.
(b) Calculate the standard deviation of the distribution.
14
12
10
8
6
4
2
0
0.5
[3]
[4]
Marks
10.5 20.5 30.5 40.5 50.5
Answer:
(a) 24.07 (b) 11.74
_________________________________________________________________________________________
5. SPM 2006 Paper 1 Compulsory Question No. 24
A set of positive integers consists of 2, 5 and m.
The variance for this set of integers is 14.
Find the value of m.
[4]
Answer:
11
_________________________________________________________________________________________
Statistics
10](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-10-320.jpg)
![6. SPM 2006 Paper 2 Compulsory Question No. 6
The table beside shows the frequency distribution of the scores of a group
of pupils in a game.
(a) It is given that the median score of the distribution is 42.
Calculate the value of k.
(b) Use the graph paper provided by the invigilator to answer this question.
Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to
2 pupils on the vertical axis, draw a histogram to represent the
frequency distribution of the scores.
Find the mode score.
(c) What is the mode score if the score of each pupil is increased by 5?
Score
10 - 19
20 - 29
30 - 39
40 - 49
50 - 59
60 - 69
Number
of Pupils
1
2
8
12
k
1
[4]
[1]
Answer:
(a) 4 (b) 42.6 (c) 47.6
_________________________________________________________________________________________
7. SPM 2007 Paper 1 Compulsory Question No. 22
A set of data consists of five numbers. The sum of the numbers is 60 and
the sum of the squares of the numbers is 800. Find, for the five numbers
(a) the mean,
(b) the standard deviation.
[3]
Answer:
(a) 12 (b) 4
_________________________________________________________________________________________
8. SPM 2007 Paper 2 Compulsory Question No. 5
The table below shows the cumulative frequency distribution for the scores of 32 students in a competition.
Score
< 10
< 20
< 30
< 40
< 50
Number of students
4
10
20
28
32
(a) Based on the table above, copy and complete the table below.
Score
0-9
10 - 19
20 - 29
Number of students
[1]
30 - 39
(b) Without drawing an ogive, find the interquartile range of the distribution.
Answer:
(a) 4, 6, 10, 8, 4
(b)
40 - 49
[5]
22
3
_________________________________________________________________________________________
9. SPM 2008 Paper 1 Compulsory Question No. 22
A set of seven numbers has a mean of 9.
(a) Find x.
(b) When a number k is added to this set, the new mean is 8.5. Find the value of k.
[3]
Answer:
(a) 63 (b) 5
_________________________________________________________________________________________
10. SPM 2008 Paper 2 Compulsory Question No. 6
Marks Number of Candidates
The table beside shows the marks
10 - 19
1
obtained by 40 candidates in a test.
20 - 29
x
Given that the median mark is 35.5,
30 - 39
y
find the value of x and of y.
40 - 49
10
Hence, state the modal class.
[6]
50 - 59
8
Answer:
x = 13, y = 5, modal class = 20 - 29
________________________________________________________________________________________
Statistics
11
[3]](https://izqule7twkl7vq3ljkxejyz-s-a2157.bj.tsgdht.cn/statistics-140118081812-phpapp01/85/Statistics-11-320.jpg)
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Statistics
- 1. STATISTICS 1. Measures of Central Tendency Mode, median and mean For a sample of discrete data, the mode is the observation, x with the highest frequency, f. 1 2N −F For grouped data in a cumulative frequency table, the median = L + C fm where L is the lower boundary of the median class N is total frequency F is the cumulative frequency before the median class C is the interval of the median class fm is the frequency of the median class ∑ x . For grouped data, x = The mean is the average of all the observations. Hence, x = N (a) Find the mode, median and mean for 2, 3, 1, 2, 6, 8, 9, 3, 2, 3. (b) [Mode = 2, Median = 3, Mean = 4] Find the modal class and calculate the median (d) and mean for the data in the table below. Points frequency 0-4 2 5-9 5 2 1 4 3 6 2 9 12 13 1 2 1 [Mode = 4, Median = 6, Mean = 7.2] Find the modal class and calculate the median and mean for the data in the table below. Marks 10 - 14 15 - 19 8 3 . Find the mode, median and mean for the data in the table below. Score Frequency (c) ∑f x ∑f f 1 - 15 16 - 30 31 - 45 46 - 60 61 - 75 76 - 90 8 11 25 34 16 6 [Modal class = 10-14, Median = 10.75, Mean = 10.33] [Modal class = 46-60, Median = 48.15, Mean = 46.53] Statistics 1
- 2. (e) Estimate the mode and calculate the median and mean for the histogram below. (f) [Mode = 72.8, Median = 72.71, Mean = 72.6] Statistics Estimate the mode and calculate the median and mean for the histogram below. [Mode = 152.5, Median = 150.09, Mean = 149.12] 2
- 3. (g) Find the modal class and find the median and mean for the ogive below. (h) [Modal class = 15-19, Median = 17.6, Mean = 18.625] Statistics Find the modal class and find the median and mean for the ogive below. [Modal class = 45-49, Median = 49.5, Mean = 50.25] 3
- 4. 2. Measures of Dispersion: Range, quartiles, interquartile range, variance and standard deviation. The range of a sample is the difference between the observations with the highest value and the lowest value. 1 1 4N −F 4N −F The first quartile, Q1 = LQ1 + CQ1 and the third quartile, Q3 = LQ3 + CQ3 f Q1 f Q3 Interquartile range = Q3 – Q1. It should be recognised that the median is Q2. ∑ (x − x ) = 2 The variance, σ 2 N ∑x = 2 2 − x for discrete data. N ∑ ( f x − x) = ∑f 2 For grouped data, σ 2 ∑ (x −x N ) 2 = ∑ ( f x − x) ∑f ∑x (a) (c) 2 −x . 2 N 2 For grouped data, σ = 2 variance = σ The standard deviation is actually For discrete data, σ = ∑f x ∑f = = 2 −x . ∑f x ∑f 2 2 −x . Find the range and the interquartile for 5, 1, 2, 3, 4, 6, 3, 8, 2, 5, 9. (b) [Range = 8, Interquartile range = 4] Find the range and the interquartile for (d) Score 1 4 5 6 8 9 Frequency 1 3 1 1 2 1 [Range = 8, Interquartile range = 4] Statistics Find the range and the interquartile range for 12, 17, 13, 19, 15, 8, 12, 11. [Range = 11, Interquartile range = 4.5] Find the range and the interquartile range for Points 2 4 6 8 No. of person 3 5 2 2 [Range = 6, Interquartile range = 3] 4
- 5. (e) Find the interquartile range for the table below. Marks No. of Students 1 - 20 21 - 40 41 - 60 61- 80 4 9 12 10 (f) Find the interquartile range for the table below. Age (year) 1-20 21-40 41-60 61-80 No. of residents 66 99 57 28 81 - 100 5 [Interquartile range = 40.77 marks] (e) Calculate the interquartile range for the ogive below. (f) [Interquartile range = 12.5 mm] Statistics [Interquartile range = 28.95 years] Calculate the interquartile range for the ogive below. [Interquartile range = 6.5 years] 5
- 6. (g) Calculate the interquartile range for the histogram below. (h) [Interquartile range = 5.295 kg] Statistics Calculate the interquartile range for the histogram below. [Interquartile range = 18.03 minutes] 6
- 7. (a) Find the mean, variance and the standard deviation for the data below. 5, 12, 6, 3, 6, 10. (b) Find the mean, variance and the standard deviation for the data below. 18, 12, 16, 11, 19, 18, 12, 14. [Mean = 15, 92 = 7.5, 9 = 2.739] Complete the table below and calculate the mean, variance and the standard deviation for the data. x 2 4 6 8 10 12 f 1 2 2 2 1 2 fx fx x−x x−x (x − x )2 2 f (x − x ) (c) [Mean = 7, 92 = 9.333, 9 = 3.055] Complete the table below and calculate the (d) mean, variance and the standard deviation for the data. x 1 2 3 4 5 6 f 1 3 4 7 3 2 (x − x )2 2 f (x − x ) [Mean = 7, 92 = 9.333, 9 = 3.055] Statistics [Mean = 7.2, 92 = 10.66, 9 = 3.265] 7
- 8. (e) Complete the table below and calculate the mean, variance and the standard deviation for the data. Class 1 - 3 4 - 6 7 - 9 10 - 12 13 - 15 f 1 3 8 6 2 (f) Complete the table below and calculate the mean, variance and the standard deviation for the data. Class 10 - 19 20 - 29 30 - 39 40 - 49 f 5 7 5 3 x x fx fx x2 x2 fx 2 fx 2 [Mean = 8.75, 92 = 8.8875, 9 = 2.981] [Mean = 27.5, 92 = 101, 9 = 10.05] (g) Complete the table below and calculate the (h) Complete the table below and calculate the mean, variance and the standard deviation for mean, variance and the standard deviation for the data. the data. Class 1 - 5 6 - 10 11 - 15 16 - 20 21 - 25 Class 0 - 19 20 - 39 40 - 59 60 - 79 f 3 13 23 9 2 f 4 7 6 3 x x fx fx 2 x2 fx 2 fx 2 x [Mean = 12.4, 92 = 20.64, 9 = 4.543] Statistics [Mean = 37.5, 92 = 376, 9 = 19.39] 8
- 9. (i) (j) Extract the data from the histogram above and complete the table below. Calculate the mean, variance and the standard deviation for the data. Extract the data from the histogram above and complete the table below. Calculate the mean, variance and the standard deviation for the data. Class x f fx fx 2 Class [Mean = 55.55, 92 = 41.6475, 9 = 6.453] Statistics x f fx fx 2 [Mean = 150.3, 92 = 172.36, 9 = 13.13] 9
- 10. 1. SPM 2003 Paper 2 Compulsory Question No. 5 A set of test marks: x1, x2, x3, x4, x5, x6 has a mean value of 5 and standard deviation of 1.5. (a) Find the (i) sum of marks, x, (ii) sum of squared marks, x2. (b) Each mark is multiplied by 2 and then 3 is added. For the set of new marks, find the (i) mean, (ii) variance. [3] [4] Answer: (a) (i) 30 (ii) 163.50 (b) (i) 13 (ii) 3 _________________________________________________________________________________________ 2. SPM 2004 Paper 2 Compulsory Question No. 4 A set of data contains 10 numbers. The sum of these numbers is 150 and the sum of their squares is 2472. (a) Find the mean and variance for these 10 numbers. (b) Another number is added to this set of data and the mean increase by 1. Find (i) the value of this number, (ii) the standard deviation for the set of 11 numbers. [3] [4] Answer: (a) mean = 15, variance = 22.2 (b) (i) 26 (ii) 5.494 _________________________________________________________________________________________ 3. SPM 2005 Paper 1 Compulsory Question No. 23 The mean of four numbers is m . The sum of the squares of the numbers is 100 and the standard deviation is 3k. Express m in terms of k. [3] Answer: 25 - 9k2 _________________________________________________________________________________________ 4. SPM 2005 Paper 2 Compulsory Question No. 4 Number of Pupils The diagram beside is a histogram which represents the distribution of the marks obtained by 40 pupils in a test. (a) Without using an ogive, calculate the median mark. (b) Calculate the standard deviation of the distribution. 14 12 10 8 6 4 2 0 0.5 [3] [4] Marks 10.5 20.5 30.5 40.5 50.5 Answer: (a) 24.07 (b) 11.74 _________________________________________________________________________________________ 5. SPM 2006 Paper 1 Compulsory Question No. 24 A set of positive integers consists of 2, 5 and m. The variance for this set of integers is 14. Find the value of m. [4] Answer: 11 _________________________________________________________________________________________ Statistics 10
- 11. 6. SPM 2006 Paper 2 Compulsory Question No. 6 The table beside shows the frequency distribution of the scores of a group of pupils in a game. (a) It is given that the median score of the distribution is 42. Calculate the value of k. (b) Use the graph paper provided by the invigilator to answer this question. Using a scale of 2 cm to 10 scores on the horizontal axis and 2 cm to 2 pupils on the vertical axis, draw a histogram to represent the frequency distribution of the scores. Find the mode score. (c) What is the mode score if the score of each pupil is increased by 5? Score 10 - 19 20 - 29 30 - 39 40 - 49 50 - 59 60 - 69 Number of Pupils 1 2 8 12 k 1 [4] [1] Answer: (a) 4 (b) 42.6 (c) 47.6 _________________________________________________________________________________________ 7. SPM 2007 Paper 1 Compulsory Question No. 22 A set of data consists of five numbers. The sum of the numbers is 60 and the sum of the squares of the numbers is 800. Find, for the five numbers (a) the mean, (b) the standard deviation. [3] Answer: (a) 12 (b) 4 _________________________________________________________________________________________ 8. SPM 2007 Paper 2 Compulsory Question No. 5 The table below shows the cumulative frequency distribution for the scores of 32 students in a competition. Score < 10 < 20 < 30 < 40 < 50 Number of students 4 10 20 28 32 (a) Based on the table above, copy and complete the table below. Score 0-9 10 - 19 20 - 29 Number of students [1] 30 - 39 (b) Without drawing an ogive, find the interquartile range of the distribution. Answer: (a) 4, 6, 10, 8, 4 (b) 40 - 49 [5] 22 3 _________________________________________________________________________________________ 9. SPM 2008 Paper 1 Compulsory Question No. 22 A set of seven numbers has a mean of 9. (a) Find x. (b) When a number k is added to this set, the new mean is 8.5. Find the value of k. [3] Answer: (a) 63 (b) 5 _________________________________________________________________________________________ 10. SPM 2008 Paper 2 Compulsory Question No. 6 Marks Number of Candidates The table beside shows the marks 10 - 19 1 obtained by 40 candidates in a test. 20 - 29 x Given that the median mark is 35.5, 30 - 39 y find the value of x and of y. 40 - 49 10 Hence, state the modal class. [6] 50 - 59 8 Answer: x = 13, y = 5, modal class = 20 - 29 ________________________________________________________________________________________ Statistics 11 [3]