FORM SIX MATHEMATICS STUDY NOTES STATISTICS 1
UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU:0787237719
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FORM SIX MATHEMATICS STUDY NOTES STATISTICS 1
Is the branch of Mathematics which deals with the collection, presentation and analysis of data obtained from various experiments.
FREQUENCY DISTRIBUTION
Is the arranged data summarized by distributing it into classes or categories with their frequency
Example;
| Variables | 20 | 40 | 50 | 60 | 70 |
| Frequency | 2 | 7 | 5 | 4 | 3 |
GRAPHICAL REPRESENTATION
Is often useful to represent frequency distribution by means of diagrams.There are different types of diagrams. These are;
1. Line graph
2. Cumulative frequency curve (Ogive)
3. Circles or Pie
4. Bar chart (Histogram)
5. Frequency polygon
TYPES OF DATA
They are;
i) Ungrouped.
ii) Grouped data.
UNGROUPED DATA
Is the type of data in which each value is taken separately which represent to each other e.g. 20, 30 ,40 ,50 etc
REPRESENTATION OF UNGROUPED DATA
-Ungrouped data can be represented by;
(a) Frequency distribution table
(b) Cumulative frequency distribution table
(c) Frequency histogram
(d) Frequency polygon
(e) Frequency curve
(f) Cumulative frequency distribution curve (O gives)
A.FREQUENCY DISTRIBUTION TABLE
Is the table of values with their corresponding frequencies
For instance;
| Values | 20 | 30 | 40 | 50 | 60 | 70 |
| frequency | 2 | 3 | 5 | 6 | 2 | 1 |
B. CUMULATIVE FREQUENCY TABLE
Is the table of values with their corresponding cumulative frequencies
For instance;
| Values | 20 | 30 | 40 | 50 | 60 | 70 |
| frequency | 2 | 5 | 10 | 16 | 18 | 19 |
C. FREQUENCY HISTOGRAM
Is the graph which is drawn by using frequency against given values.
Example
D.FREQUENCY POLYGON
Is the polygon which is drawn by using the corresponding points of frequencies against a given value.
Example
E. THE FREQUENCY CURVE
Is the curve which is drawn by joining (free hand) the corresponding points of frequency against given values.
Example
F.CUMULATIVE FREQUENCY CURVE (O GIVE)
Is the curve of cumulative frequency against given values
Example
MEASURES OF CENTRAL TENDENCY
These are;
I) Mean
ii) Mode
iii) Median
iv) Harmonic mean
v) Geometric mean
1. MEAN
Is obtained by adding together all the items and dividing by the number of items.
Mean () =
2. MODE
Is the number (value) which occur most frequently
For instance
i) Given 2,3,5,5,6,7,7,7,9
→7 has a frequency of 3
→Hence 7 is the mode
ii) Given 1,2,3,4,5,5,6,7,
→then 4 and 5 are the mode their frequency is 2
iii) Given 2, 33, 4, 5, 6, 7= there is no mode at all
3. MEDIAN
Is the middle number (value) when the data is arranged in the order of site.
N.B
I) When the total number of items is ODD say “N” the value of
Items give the mode.
II) When the total number of items is EVEN, say “N” there are two middle items, then the mean of the values of ½ Nth and (½ N+1)th item is the median
4. HARMONIC MEAN
Is the reciprocal of arithmetic mean of their values.
– If it is harmonic mean
Then
For value of x1, x2 ,x3…xn with the frequency, f1,f2, f3+…….. fn respectively
The harmonic mean is given by
5. GEOMETRIC MEAN
For the values of x1, x2, x3, x4,………xn then the geometric mean is given by
Examples
Find the mean of 20, 22, 25, 28, and 30
Solution
Given 20, 22, 25, 28, 30
∑f =5
Mean () =
() =25
2. Find the mean of the following;
| No | 8 | 10 | 15 | 20 |
| f | 5 | 8 | 8 | 4 |
Solution
Consider the table below;
| No | F | Fx |
| 8 | 5 | 40 |
| 10 | 8 | 80 |
| 15 | 8 | 120 |
| 20 | 4 | 80 |
∑f=25 ∑ f(x)= 320
Mean () =
3. Find the median of 6, 8, 9, 10, 11, 12 and 13.
Solution
Given 6, 8, 9, 10, 11, 12, 13
N=7
Position =
=
= 4th
Median is 10
4. Find the mode of the following items
0,1,6,7,2,3, 7,6,6,2,6,0, 5, 6, 0
Solution
From the given data 6 has a frequency of 5
→6 is the mode
5. Find the geometric mean of 4,16,8
Solution
Number 4,8,16
N=3
From
G= 8
→Geometric mean is 8
6. Calculate the harmonic mean of the data 4,8,16
Solution
Numbers4, 8, 16
N=3
H = 6.857
The harmonic mean =6.857
B.MEASURES OF DISPERSION (Variability)
These are;
i) Variance
ii) Standard deviation
iii) Mean deviation
1. VARIANCE
This is given by;
The formula is used for ungrouped data
Also,
Var(x) =
2. STANDARD DEVIATION
This is given by;
S.D =
S.D=
S.D
3. MEAN DEVIATION
This is given by
M.D =
Also f =1
M.D =
Examples
7. From the distribution 1, 2, 3, 4, 5 find variance
8. Given the distribution 2, 3, 4, 5, 6, 7, 4, 5, 3 find
a) Variance
b) Standard deviation
c) Mean deviation
Solution
Consider the distribution table
Variance
Form
Var(x) = 2.223.
b) Standard deviation
S.D =
S.D =
S.D = 1.491
c). Mean deviation
=
= 1.25556
C.MEASURES OF POSITION
These are;
i) Quartile
ii) Decile
iii) Percentile
1. QUARTILE
This is the division of frequency distribution into four equal parts.
Hence there are;
1st –quartile
2nd-quartile
3rd-quartile
-The position of 1st quartile is
-The position of 2nd quartile is
-The position of 3rd quartile is
N.B. 2nd quartile is the median
INTER-QUARTILE RANGE
This is the different between 1st and 3rd quartile
Mathematically;
Inter-quartile =3rd quartile – 1st quartile
I.Q.R = Q3 – Q1
SEMI- INTER QUARTILE RANGE
This is given as a half of inter-quartile range
i.e. semi inter quartile range
Semi I.Q.R=
2. DECILE
This is the division of frequency distribution into ten equal parts
Hence there are
1st Decile
2nd Decile
3rd Decile
5th Decile
9th Decile
The position of 1st decile is
The position of 2nd decile is
The position of 3rddecile is
The position of 5rd decile is
The position of 9th decile is
N.B
The 5th decile is the median
PERCENTILE
This is the division of frequency distribution into 100, equal parts, hence these are;
2nd percentile
3rd percentile
4th percentile
50th percentile
99th percentile
The position of 1st percentile is
The position of 2nd percentile is
The position of 3rd percentile is
The position of 50th percentile is
The position of 99th percentile is
N.B. 50th Percentile is the median
Examples
Given the distribution 2, 3, 4, 5, 6, 7, 8, 9 find
i) 1st quartile
ii) 2nd quartile
iii) 3rd quartile
iv) Inter- quartile range
v) Semi inter-quartile range
Solution
Given 2, 3, 4, 5, 6, 7, 8, 9
n=8
i) 1st quartile (Q1)
Position of (Q1) =
=2nd Value
1st quartile =
ii) second quartile (Q2)
Position of Q2 =
= 4th value
2nd quartile =
iii)Third quartile (Q3)
Position of Q3 =
3rd quartile =
iv) Inter-quartile range
I.Q.R = Q3-Q1
7.5 3.5 = 4
v) Semi inter-quartile range
S.I.QR
QUESTIONS
3. Given the distribution 1, 2, 3, 4, 5, 6, 7, 8, 9 find
i) Quartile 1
ii) Quartile 2
iii). Quartile 3
4. Given the distribution 2, 3, 5, 6, 7, 8, 9, 9, 10, 11, 12 find.
a) Lower quartile
b) Median
c) Upper quartile
5. From the distribution 20, 23, 23, 26, 27, 28 find
i) Q1 ii)Q2 iii)Q3
6. Given the distribution 147,150,154,158,159,162,164,165 find i) Q1 ii) Q2 iii) Q3
7. Given the frequency distribution 10,12,13,15,17,19,24,26,26, find i)Q1 ii) Q2 iii) Q3
8. From the frequency distribution 2,3,4,5,6,7,8,9,10,11,12
Find i) first decile
ii) 4th decile
iii) 5th decile
9. The following table below shows the scores obtained when a dice thrown 60 times .Find the median score
| SCORE | 1 | 2 | 3 | 4 | 5 | 6 |
| FREQUENCY(F) | 12 | 9 | 8 | 13 | 9 | 9 |
10. Find the median and inter quartile range of the frequency distribution
| X | 5 | 6 | 7 | 8 | 9 | 10 |
| F | 6 | 11 | 15 | 18 | 16 | 5 |
GROUPED DATA
This is the grouping of frequency distribution obtained from various experiments.0-9, 10-19, 20-29, 30,-39
TERMINOLOGIES
1 .LOWER CLASS INTERVAL
This is the lower interval or lower limit of the class.
Example
x1 →x2, x5→ x6, x7 →x8
x1, x5, x7 are lower interval of the classes
2. UPPER CLASS INTERVAL
This is the upper interval of upper limit of the class.
E.g. x1 →x2, x3→ x4, x5 →x6
x2 ,x 4 and x6 are the upper interval of the classes
3. CLASS MARK(X)
Is the average between class intervals.
Mathematically;
Where L.C.L=lower class limit
U.C.L= upper class limit
4. CLASS BOUNDARIES
These are real limits of the classes
a) Upper class boundaries
i.e.
Where;
U.C.B = upper class boundary
U.C.L = Upper class limit
b) Lower class boundary
L.C.B = L.C.L-0.5
Where;
L.C.B Lower class boundary
L.C.L = Lower class limit
5. CLASS SIZE(WIDTH)
Is the difference between lower or upper and upper class limit or class boundaries between two successive classes.
Mathematically;
REPRESENTATION OF GROUPED DATA
These are;
i) Frequency distribution table
ii) cumulative frequency distribution table
iii) frequency histogram
iv)Cumulative frequency curve (o give)
v) frequency polygon
vi) scatter diagram
1. FREQUENCY DISTRIBUTION TABLE
Is the table of class interval with their corresponding frequency.
| class interval | 0-9 | 10-19 | 20-29 | 30-39 |
| frequency | 2 | 7 | 3 | 4 |
How to prepare frequency distribution table
A. IF CERTAIN CLASS INTERVAL PROVIDED
In this case, prepare the frequency table by using interval provided
Complete the column of frequency by random inspection.
→The last interval should end up to where the highest value belong to
FORM SIX MATHEMATICS STUDY NOTES STATISTICS 1
B. IF CERTAIN CLASS MARKS PROVIDED
1st method
i) Mark the lowest value from the data provided and call it as lower limit (L) of the first class interval
ii)using the formula of class mark calculate the upper limit(u)
Class mark
iii) Indicate the first interval as L to U
The proceed under the same interval in order to get frequency distribution table
2nd method
i) Get class size (c) then find the value of (
ii) By using the value of ( get the intervals as follows
Lower interval
Upper interval
iii) By using class interval obtained prepare the frequency distribution table
C. IF CLASS MARK AND CLASS INTERVALS ARE NOT PROVIDED
In this case the frequency distribution table should be prepared under the following steps.
i) Perform random inspection of highest and lowest value
ii) Determine the value of the range as follows
Range (R) =
R
iii) Determine the class size according to required number of classes by using the formula
Where r=range
C=class size
N=number of class required
E.g. C=
C=
iv) By using class size obtained above, prepare frequency distribution table by regarding condition existing from the data
Examples
1.The following are the results from mathematical test of 20 students 27,21,24,27,31,40,45,46,50,48,38,29,49,98,35,34,44,23,25,49 prepare the following distribution table by using class interval 21-25,26-30,31-35
2. The following are the results of physics test of 50 students at Azania sec school 21,23,48,54,64,77,68,52,31,40,33,43,53,61,71.82,75,61,64,34,25,26,31,32,36,48,45,44,55,52,60,67,67,7,74,78,80,85,90,97,26,27,37,38,34,39,40.41.45.48 prepare the frequency distribution table using class mark 23,28,33
3. 3 .The data below give time in minutes .it takes a computer to drive to work for a period of lasting 50 days
25,40,27,43,23,28,39,33,29,26,34,32,28,30,39,32,25,27,28,28,27,35,28,46,24,24,22,31,28,27,35,28,46,24,24,22,31,28,27,31,23,32,36,22,26,34,30,27,25,42,25,37,30.27,31, 30, 48, 28, 24
Construct a frequency distribution table having six classes for which 20 is the lowest unit of the first class and 49 is the upper limit of the size of class.
Solution 1
Frequency distribution table
| CLASS INTERVAL | FREQUENCY |
| 21-25 | 4 |
| 26-30 | 3 |
| 31-35 | 3 |
| 36-40 | 2 |
| 41-45 | 2 |
| 46-50 | 6 |
Solution 2
Given:
Class mark(x) = 23, 28, 33
Class size (i) 28-23=5
→1st class interval 23 – 2 = 21
=21
Upper limit 23 + 2 = 25
=25
= 21→25
→2nd class interval
Lower limit = 28 – 2
=26
Upper limit = 28 + 2
=30
= 26→30
→3rd class interval
Lower limit = 33 – 2
= 31
Upper limit = 33 + 2
=35
Others36-40
41-45
46-50
Frequency distribution table
| Class interval | Frequency |
| 21-25 | 3 |
| 26-30 | 3 |
| 31-35 | 5 |
| 36-40 | 6 |
| 41-45 | 5 |
| 46-50 | 4 |
| 51-55 | 4 |
| 56-60 | 2 |
| 61-65 | 4 |
| 66-70 | 4 |
| 71-75 | 4 |
| 76-80 | 3 |
| 81-85 | 2 |
| 86-90 | 1 |
Solution 3
Range(R) = H – C
= 49 – 20
= 29
Also
N=number of class required
N=6
C=
C =
1st class interval
C= U.C.B – L.C.D
C = (U.C.L+0.5) (L.C.L-0.5)
C = U.C.L – L.C.L + 1
5 = U.C.L – 20 + 1
5 = U.C.L – 19
U.C.L = 24
→20-24
Other intervals are 25→29, 30→34, 35→39, 40→44, 45→49
Frequency distribution table
| Class interval | Frequency |
| 20-24 | 8 |
| 25-29 | 20 |
| 30-34 | 12 |
| 35-39 | 5 |
| 40-44 | 3 |
| 45-49 | 2 |
II. CUMULATIVE FREQUENCY DISTRIBUTION TABLE
Is the table of class interval with their corresponding Cumulative frequency
Example
| Class interval | 0-9 | 10-19 | 20-29 |
| Cum freq | 3 | 7 | 20 |
III. FREQUENCY HISTOGRAM
Is the graph which is drawn by using frequencies against class mark
Example
IV) FREQUENCY POLYGON
Is the polygon which is drawn by joining the corresponding points of frequencies against the class marks.
Example;
Class mark
Frequency
V) CUMULATIVE FREQUENCY CURVE (O GIVE)
Is the curve which is drawn by joining (free hand) the corresponding points of cumulative frequencies against the upper class boundary.
Example;
VI) SCATTERED DIAGRAM
Is the diagram where frequencies scattered against class mark without connecting the points.
MEASURES OF GROUPED DATA
These are;
a) Measures of central tendency
b) Measures of dispersion (variability)
c) Measures of position
A.MEASURES OF CENTRAL TENDENCY
These are i) mean
ii) Median
iii) Mode
I. MEAN ()
-By direct method
Mean ( )
Where
∑f(x) → is the sum of frequencies times class mark
∑f →sum of frequency
| X | X1 | X2 | X3 | Xn |
| F | F1 | F2 | F3 | Fn |
By assumed mean method
I.e. mean(x) =
Where
X → is the class mark
A → is the assumed mean
D → deviation
Substitute (ii) into (i)
BY CODING METHOD
Where;
d → is the deviation
C → is the class size
n → is the coding number
Substitute (ii) into (i)
Examples
1. From the following distribution
| X | 10 | 20 | 30 | 40 | 50 |
| F | 16 | 18 | 25 | 19 | 22 |
Find the mean by
i) Direct method
ii) assumed mean
iii) Coding method
Solution
Consider the distribution table
| X | F | F(x) | d= x – a | f(d) | u=
|
fu |
| 10 | 16 | 160 | -20 | -320 | -2 | -32 |
| 20 | 18 | 360 | -10 | -180 | -1 | -18 |
| 30 | 25 | 750 | 0 | 0 | 0 | 0 |
| 40 | 19 | 160 | 10 | 190 | 1 | 19 |
| 50 | 22 | 1100 | 20 | 440 | 2 | 44 |
 |
 |
 |
 |
Let A = Assumed mean
= 30
C =Class size
= 10
a) By direct mean method
I.e. Mean ( )
Mean (x) = 25.30
b) By assumed mean method
c) By coding method
MODE
This is the value which occurs most frequently in grouped data mode can be determined by using two methods
a) By estimation from histogram
b) By calculation method
A) MODE FROM HISTOGRAM
-Consider the three bars under consideration of the highest bar with their two adjacent bars from the histogram.
BY CALCULATION METHOD
-Assume that the figure below represents three rectangles of the histogram of the frequency distribution of central rectangle corresponding to modal class.
Where
D1 → is the difference between the frequencies of the mode class and the frequency of the class just before the modal class.
D2 → is the difference between the frequency of the modal class and the frequency of the class just after the modal class
Lc → is the lower class boundary of the modal class
U → is the upper class boundary of the modal class
Hence from the histogram
D2(X – LC) = D1 (U – X)
D2 X – D2LC =D1U – D1X
D1X + D2X = D2LC + D1U
X (D1 + D2) = D2 LC + D1U
But
U LC = class size(C)
C= U LC
U = C + LC
X(D1 + D2 ) = D2LC + D1 (LC+ C)
2. MEDIAN
Position of the median class N/2 hence therefore using interpolation
Where;
LC→ is lower boundary of the median class
→ Sum of the frequency before the median class
fm→ frequency of the modal class
c → class size
Example
1. from the following distribution table
| Class interval | Frequency | Comm. freq |
| 1-7 | 8 | 8 |
| 8-14 | 10 | 18 |
| 15-21 | 22 | 40 |
| 22-28 | 15 | 55 |
| 29-35 | 7 | 62 |
| 36-42 | 18 | 80 |
Find i) mode
ii) Median
2. Find the mean and the median o the following distribution
| Class interval | Frequency |
| 0.20-0.24 | 6 |
| 0.25-0.29 | 12 |
| 0.30-0.34 | 19 |
| 0.35-0.39 | 13 |
3. Given the frequency distribution table below
| Class interval | Frequency |
| 16.50-16.59 | 25 |
| 16.60-16.69 | 47 |
| 16.70-16.79 | 65 |
| 16.80-16.89 | 47 |
| 16.90-16.99 | 16 |
Solution 1
Consider the distribution table
| Class interval | Frequency | Comm. Freq |
| 1-7 | 8 | 8 |
| 8-14 | 10 | 18 |
| 15-21 | 22 | 40 |
| 22-28 | 15 | 55 |
| 29-35 | 7 | 62 |
| 36-42 | 18 | 80 |
i) mode from the table
Modal class = 15→21
ii) median
Position of median class
=
=40
Median class =15→21
LC= (15 – 0.5)
= 14.5
∑fb=18
Fm=22
C=7
Then
Median = 21.5
MEASURES OF DISPERSION (variability)
These are
i) Variance
ii) standard deviation
VARIANCE
→Variance by direct mean method
Recall
VARIANCE BY ASSUMED MEAN METHOD
Recall
Put X – A= d
X = A + d…… (ii)
Where
X → class mark
A → Assumed mean
Substitute……..ii)………..i) as follows
VARIANCE BY CODING METHOD
Recall,
Where;
D = deviation
C = class size
U = coding number
Substitute (ii) into (i)
2. STANDARD DEVIATION
This is given by
→By direct mean method
→By assumed mean method
→By Coding method
questions
1. Given the distribution class interval frequency
| Class interval | Frequency |
| 1-5 | 8 |
| 6-10 | 18 |
| 11-15 | 9 |
| 16-20 | 25 |
| 21-25 | 40 |
Find the standard deviation by coding method.
2. The table below shows the frequency distribution of intelligence quotient (IQ)of 500 individuals
| I.Q | Frequency |
| 82-85 | 5 |
| 86-89 | 19 |
| 90-93 | 32 |
| 94-97 | 49 |
| 98-101 | 71 |
| 102-105 | 92 |
| 106-109 | 75 |
| 110-113 | 56 |
| 114-117 | 39 |
| 118-121 | 28 |
| 122-125 | 18 |
| 126-129 | 10 |
| 130-133 | 6 |
Using coding method find
i) Mean
ii) Standard deviation
c. MEASURES OF POSITION
These are
1.quartile
2. deciles
3. Percentile
1.QUARTILE
Recall
2. DECILE
Recall
3. PERCENTILE
Recall
Note
2nd quartile 5th, decile and 50th percentile are MEDIANS
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