Download - Project Ahmedawad
Frequency Distribution
1.1 Raw Data
1.2 Arrays
1.3 Frequency distribution
1.4 Histogram and Frequency Polygons
1.5 The relative Histogram and Frequency Polygons
1.6 Cumulative frequency and Ogives
1.7 Relative Cumulative frequency and percentage
Ogives
1.8 Frequency Curve and Smoothed Ogives
Raw Data Are collected data that have not been organized numerically
205 149 139 132 159 130 158 148 142 147 176 158 140 154 135 131 172 152 150 190 175 182 145 146 175 184 143 135 195 173 143 132 186 161 160 129 144 142 146 141 136 138 179 149 139 190 171 135
Data Collector Signature Date
Ahmed Ahmed 10/04/2011
Check sheet Part Name: tubes Order No: 68
Part No: 010 Factory Name: Coca Cola
Dimension: diameter Production Line: 1
Unit of measure: mm Section: Turning
Total No: 48 Manufacturing Stage: Final
Lot No: 275 Remarks: All bottles inspected
An Array
An array is an arrangement of raw numerical data in ascending or descending
order of magnitude
Ascending array
Data Collector Signature Date
Ahmed Ahmed 10/04/2011
Check sheet Part Name: tubes Order No: 68
Part No: 010 Factory Name: Coca Cola
Dimension: diameter Production Line: 1
Unit of measure: mm Section: Turning
Total No: 48 Manufacturing Stage: Final
Lot No: 275 Remarks: All bottles inspected
135 135 135 132 132 131 130 129 142 142 141 140 139 139 138 136 148 147 146 146 145 144 143 143 159 158 158 154 152 150 149 149 176 175 175 173 172 171 161 160 205 195 190 190 186 184 182 179
Descending array
205 195 190 190 186 184 182 179 176 175 175 173 172 171 161 160 159 158 158 154 152 150 149 149 148 147 146 146 145 144 143 143 142 142 141 140 139 139 138 136 135 135 135 132 132 131 130 129
Data Collector Signature Date
Ahmed Ahmed 10/04/2011
Check sheet Part Name: tubes Order No: 68
Part No: 010 Factory Name: Coca Cola
Dimension: diameter Production Line: 1
Unit of measure: mm Section: Turning
Total No: 48 Manufacturing Stage: Final
Lot No: 275 Remarks: All bottles inspected
Frequency Distributions
General rule for forming frequency distribution :
1- Determine the largest and smallest numbers in the raw data and
thus find the range
Range (R) = 205 – 129 = 76
2- Divide the range into a convenient number of class (n)
Number of class interval is chosen (n) = 7
The class interval (c) = 76 / 7 =10.8
3- Determine the number of observations failing into each class
interval , using tally
Class boundary Class limit tally Frequency
128.5-139.5 129-139 //// //// // 12 139.5-150.5 140-150 //// //// //// 15 150.5-161.5 151-161 //// / 7 161.5-172.5 162-172 // 2 172.5-183.5 173-183 //// / 6 183.5-194.5 184-194 //// 4 194.5-205.5 195-205 // 2
Table 1.1 Detailed Frequency Distribution of Diameter of tubes in mm
Class boundary =
Histogram and Frequency Polygons
Histogram and frequency polygon are two graphic representations of frequency
distribution.
Histogram : consists of rectangles, frequency polygon is a line graph connecting the
midpoints of the tops of histogram.
Class limit Frequency( f)
129-139 12
140-150 15
151-161 7
162-172 2
173-183 6
184-194 4
195-205 2
Total 48
(Figure 1-1) The Histogram and Frequency Polygons of diameters of tubes
0
2
4
6
8
10
12
14
16
123 134 145 156 167 178 189 200 211
No
. of
tub
es
(Fre
qu
en
cy)
tubes diamter in mm (classes)
The relative Histogram and Frequency Polygons The relative of class is the frequency of the class divided by the total frequency
of all classes
Diameter of tubes (mm)
(Class)
No of tubes / Total no. of tubes Relative frequency
129-139 25%
140-150 31%
151-161 15%
162-172 4%
173-183 13%
184-194 8%
195-205 4%
total 100%
Table 1.2 relative Frequency Distribution of Diameter of tubes in mm
(Figure 1-2) The Histogram and Frequency Polygons of diameters of tubes
0%
5%
10%
15%
20%
25%
30%
35%
123 134 145 156 167 178 189 200 211
No
of
tub
es /
To
tal n
o. o
f tu
bes
Rel
ativ
e fr
equ
ency
tubes diamter in mm (classes)
Cumulative frequency and Ogives - Less Than Method
lower than or equal to the upper boundaries of each class interval
Diameter of tubes (mm)
No of tubes Cumulative frequency
Less than 128.5 12.00
Less than 139.5 27.00
Less than 150.5 34.00
Less than 161.5 36.00
Less than 172.5 42.00
Less than 183.5 46.00
Less than 194.5 48.00 (Table 1.3) Cumulative frequency Distribution of Diameter of tubes in mm
(Figure 1-3) The Cumulative frequency Polygons of diameters of tubes
"Less Than"
0
10
20
30
40
50
60
123 134 145 156 167 178 189 200 211
Less
Th
an C
um
ula
tive
fre
qu
ency
tubes diamter in mm (classes)
- Or more
greater than or equal to the lower class boundaries of each class interval
Diameter of tubes (mm)
No of tubes Cumulative frequency
More than 128.5 12.00
More than 139.5 27.00
More than 150.5 34.00
More than 161.5 36.00
More than 172.5 42.00
More than 183.5 46.00
More than 194.5 48.00 (Table 1.4) Cumulative frequency Distribution of Diameter of tubes in mm
(Figure 1-4) The Cumulative frequency Polygons of diameters of tubes
0
10
20
30
40
50
60
134 145 156 167 178 189 200
Less
Th
an C
um
ula
tive
fre
qu
ency
tubes diamter in mm (classes)
Relative Cumulative frequency and percentage
Ogives: Is the cumulative frequency divided by the total frequency
Diameter of tubes
(mm) relative less than
Less than 128.5 25%
Less than 139.5 56%
Less than 150.5 71%
Less than 161.5 75%
Less than 172.5 88%
Less than 183.5 96%
Less than 194.5 100%
(Table 1.5) Relative Cumulative frequency Distribution of Diameter of tubes in mm "Less Than"
(Figure 1-5) The Relative Cumulative frequency Polygons of diameters of tubes " Less Than"
0%
20%
40%
60%
80%
100%
120%
123 134 145 156 167 178 189 200
Less
Th
an R
elat
ive
Cu
mu
lati
ve fr
equ
ency
tubes diamter in mm (classes)
Diameter of tubes (mm) relative or more
More than 128.5 100%
More than 139.5 96%
More than 150.5 88%
More than 161.5 75%
More than 172.5 71%
More than 183.5 56%
More than 194.5 25%
(Table 1.6) Relative Cumulative frequency Distribution of Diameter of tubes in mm "Or More"
(Figure 1-6) The Relative Cumulative frequency Polygons of diameters of tubes " Or More "
0%
20%
40%
60%
80%
100%
120%
134 145 156 167 178 189 200 211Less
Th
an R
elat
ive
Cu
mu
lati
ve fr
equ
ency
tubes diamter in mm (classes)
Frequency Curve and Smoothed Ogives: For large population broken line segments are approximated curve
(Figure 1-7) frequency Curve (Smoothed frequency Polygons)
0
2
4
6
8
10
12
14
16
0 1 2 3 4 5 6 7 8 9 10
Fre
qu
en
cy
class interval
The Arithmetic Mean The mean is the sum of the values divided by the total number of values
a) The long method
Class limit Class mark (Xj) Frequency (f) fXj
129-139 134 12 1608
140-150 145 15 2175
151-161 156 7 1092
162-172 167 2 334
173-183 178 6 1068
184-194 189 4 756
195-205 200 2 400
total 48 7433
(Table 2.1) The long method to compute the mean diameter of frequency distribution of tubes
in mm
b) The short method
i- By assuming Arithmetic Mean
Class limit Class mark (Xj) Frequency d= Xj -A fd
129-139 134 12 -33 -396
140-150 145 15 -22 -330
151-161 156 7 -11 -77
162-172 A=167 2 0 0
173-183 178 6 11 66
184-194 189 4 22 88
195-205 200 2 33 66
total 48 0 -583
(Table 2.2) The short method to compute the mean diameter of frequency distribution of tubes
in mm
i- By coding method - Used in equal size (c) - Short method for computing
Class limit Class mark (Xj) Frequency d= Xj -A u = d/c fu
129-139 134 12 -33 -3 -36
140-150 145 15 -22 -2 -30
151-161 156 7 -11 -1 -7
162-172 A=167 2 0 0 0
173-183 178 6 11 1 6
184-194 189 4 22 2 8
195-205 200 2 33 3 6
total 48 0 0 -53
(Table 2.3) The coding method to compute the mean diameter of frequency distribution of tubes
in mm
Median The median of a set of values arranged in order of magnitude (i.e., in an array) is the
middle value.
i- Analytically
Class boundary Class limit Frequency
128.5-139.5 129-139 12
139.5-150.5 140-150 15
150.5-161.5 151-161 7
161.5-172.5 162-172 2
172.5-183.5 173-183 6
183.5-194.5 184-194 4
194.5-205.5 195-205 2
total
48
ii- Graphically
Median is the value of X corresponding to the vertical line which divide the
histogram to two parts have equal area
(figure 2-1) compute the median from histogram
Represent 50% from total frequency
0
2
4
6
8
10
12
14
16
123 134 145 156 167 178 189 200 211
No
. of
tub
es
(Fre
qu
en
cy)
tubes diamter in mm (classes)
0%
20%
40%
60%
80%
100%
120%
123 134 145 156 167 178 189 200
0
2
4
6
8
10
12
14
16
123 134 145 156 167 178 189 200 211
No
. of
tub
es (
Freq
uen
cy)
tubes diamter in mm (classes)
Mode The Mode of a set of values is that value which occurs with the greatest frequency; that
is, it is the most common value
i- Analytically
Class boundary Class limit Frequency
128.5-139.5 129-139 12
139.5-150.5 140-150 15
150.5-161.5 151-161 7
161.5-172.5 162-172 2
172.5-183.5 173-183 6
183.5-194.5 184-194 4
194.5-205.5 195-205 2
total
48
i- Graphically
(figure 2-2) compute the mode from histogram
Measure of Dispersion
3.1 The Range
3.2 Standard Deviation
3.3 Variance
3.4 coefficient of variation
The Range There are two ways of defining the range for grouped data :
- Range = class mark of highest class – class mark of lowest class
Range = 200-134 = 66
- Range = upper class boundary of highest class – lower class boundary of
lowest class
Range = 205.5 – 128.5 = 77
Standard Deviation The standard deviation is the square root of the average of the squares of the
deviations from the mean
i- Long method :
- 1st formula:
o Find the arithmetic mean value
Class limit Class mark (Xj) Frequency (f) fXj
129-139 134 12 1608
140-150 145 15 2175
151-161 156 7 1092
162-172 167 2 334
173-183 178 6 1068
184-194 189 4 756
195-205 200 2 400
total 48 7433
o Find the Standard Deviation
Class limit class mark (xj) Xj - (Xj - )2 frequency f f(Xj - )
2
129-139 134 -20.854 434.8893 12 5218.672
140-150 145 -9.854 97.10132 15 1456.52
151-161 156 1.146 1.313316 7 9.193212
162-172 167 12.146 147.5253 2 295.0506
173-183 178 23.146 535.7373 6 3214.424
184-194 189 34.146 1165.949 4 4663.797
195-205 200 45.146 2038.161 2 4076.323
total 48 18933.98 (Table 3.1) The long method to compute the Standard Deviation diameter of frequency
distribution of tubes in mm (1st formula)
- 2nd
formula:
–
Class limit class mark xj X2 frequency f f Xj 2
129-139 134 17956 12 215472
140-150 145 21025 15 315375
151-161 156 24336 7 170352
162-172 167 27889 2 55778
173-183 178 31684 6 190104
184-194 189 35721 4 142884
195-205 200 40000 2 80000
total 48 1169965 (Table 3.2) The long method to compute the Standard Deviation diameter of frequency
distribution of tubes in mm (2nd formula)
–
ii- Short method :
- Assuming an arithmetic mean A
–
- Coding method
–
Assuming an arithmetic mean A
Class limit
Class mark (Xj) Frequency d=x-a fd d2 fd2
129-139 134 12 -33 -396 1089 13068
140-150 145 15 -22 -330 484 7260
151-161 156 7 -11 -77 121 847
162-172 A=167 2 0 0 0 0
173-183 178 6 11 66 121 726
184-194 189 4 22 88 484 1936
195-205 200 2 33 66 1089 2178
total 48 -583 26015
(Table 3.3) The short method to compute the Standard Deviation diameter of frequency
distribution of tubes in mm
–
Coding method
Class limit Class mark
(Xj) Frequency d=x-a u = d/c fu fu2
129-139 134 12 -33 -3 -36 108
140-150 145 15 -22 -2 -30 60
151-161 156 7 -11 -1 -7 7
162-172 A=167 2 0 0 0 0
173-183 178 6 11 1 6 6
184-194 189 4 22 2 8 16
195-205 200 2 33 3 6 18
total 48 0 0 -53 215
(Table 3.3) The coding method to compute the Standard Deviation diameter of frequency
distribution of tubes in mm
–
Variance
The variance is the average of the squares of the deviations from the mean.
Long method
i- 1st formula
–
–
ii- 2nd formula
–
–
Short method
i- Assuming an arithmetic mean A
–
–
ii- Coding method
–
–
coefficient of variation
The coefficient of variation or coefficient of dispersion is considered as an relative
dispersion
coefficient of variation ( V ) =
coefficient of variation = ( V ) =
Moments Moment of order (r) is defined as the arithmetic mean of the (r th)
power of deviations of observations
- Moments about zero :
The 1st moment about the zero is the arithmetic mean
Class limit class mark xj frequency f f Xj f Xj 2 f Xj
3 f Xj 4
129-139 134 12 1608 215472 28873248 3869015232
140-150 145 15 2175 315375 45729375 6630759375
151-161 156 7 1092 170352 26574912 4145686272
162-172 167 2 334 55778 9314926 1555592642
173-183 178 6 1068 190104 33838512 6023255136
184-194 189 4 756 142884 27005076 5103959364
195-205 200 2 400 80000 16000000 3200000000
total 48 7433 1169965 187336049 30528268021
(Table 3.1) to compute the moment about zero of frequency distribution of diameter of tubes in mm
The first moment about the origin zero ( ) :
mm
The second moment about the origin zero ( ) :
24374.27083 mm
The third moment about the origin zero ( ) :
mm
The fourth moment about the origin zero ( ) :
- Moments about mean :
Class limit
class mark Xj
frequency f f(xj- ) f(xj- )2 f(xj-
)3 f(xj- )4
129-139 134 12 -250.248 5218.672 -108830.1816 2269544.606
140-150 145 15 -147.81 1456.52 -14352.54552 141429.9835
151-161 156 7 8.022 9.193212 10.53542095 12.07359241
162-172 167 2 24.292 295.0506 3583.684976 43527.43772
173-183 178 6 138.876 3214.424 74401.0555 1722086.831
184-194 189 4 136.584 4663.797 159250.0214 5437751.23
195-205 200 2 90.292 4076.323 184029.6615 8308203.1
Total 48 0 18933.98 298092.2317 17922555.26
(Table 3.2) to compute the moment about the mean of frequency distribution of diameter of tubes in
mm
The first moment about the mean ( ) :
The second moment about the mean ( ) :
The third moment about the mean ( ) :
The fourth moment about the mean ( ) :
- Moments about origin A :
i- The rth power about any origin A :
Class limit class mark Xj frequency f f(Xj -A) f(Xj -A) 2 f(Xj -A)3 f(Xj -A) 4
129-139 134 12 -396 13068 -431244 14231052
140-150 145 15 -330 7260 -159720 3513840
151-161 156 7 -77 847 -9317 102487
162-172 A= 167 2 0 0 0 0
173-183 178 6 66 726 7986 87846
184-194 189 4 88 1936 42592 937024
195-205 200 2 66 2178 71874 2371842
Total 48 -583 26015 -477829 21244091
(Table 3.3) to compute the moment about the origin A=167 mm of frequency distribution of
diameter of tubes in mm
The first moment about the origin A ( ) :
The second moment about the origin A ( ) :
The third moment about the origin A ( ) :
The fourth moment about the origin A ( ) :
ii- The rth power about any origin A :
Class limit class mark Xj frequency f fd fd 2 fd3 fd4
129-139 134 12 -396 13068 -431244 14231052
140-150 145 15 -330 7260 -159720 3513840
151-161 156 7 -77 847 -9317 102487
162-172 A= 167 2 0 0 0 0
173-183 178 6 66 726 7986 87846
184-194 189 4 88 1936 42592 937024
195-205 200 2 66 2178 71874 2371842
Total 48 -583 26015 -477829 21244091
(Table 3.4) to compute the moment about the origin A=167 mm of frequency distribution of
diameter of tubes in mm
The first moment about the origin A=167 ( ) :
The second moment about the origin A=167 ( ) :
The third moment about the origin A=167 ( ) :
The fourth moment about the origin A=167 ( ) :
iii- The rth power about any origin A :
Class limit class mark
Xj frequency
f d= Xj -A u = d/c fu fu2 fu3 fu4
129-139 134 12 -33 -3 -36 108 -324 972
140-150 145 15 -22 -2 -30 60 -120 240
151-161 156 7 -11 -1 -7 7 -7 7
162-172 167 2 0 0 0 0 0 0
173-183 178 6 11 1 6 6 6 6
184-194 189 4 22 2 8 16 32 64
195-205 200 2 33 3 6 18 54 162
Total -53 215 -359 1451
(Table 3.5) to compute the moment about the origin A=167 mm of frequency distribution of
diameter of tubes in mm using integer u
The first moment about the origin A=167 ( ) :
The second moment about the origin A=167 ( ) :
The third moment about the origin A=167 ( ) :
The fourth moment about the origin A=167 ( ) :
Moments in Dimensionless form :
The first four moments about the mean in dimensionless form
The first moment about the mean in dimensionless form:
The second moment about the mean in dimensionless form:
The third moment about the mean in dimensionless form:
The fourth moment about the mean in dimensionless form:
Sign of skewness is determined by the sign of a3 which is +ve therefore the distribution is positively skewed. If some observation are added on the right hand side of distribution its right tail gets elongated. Fact that : Mean > Median > Mode
Kurtosis
is defined as the relative peakedness of the top of a frequency curve
one measure of kurtosis use the fourth moment about the mean expressed in
dimensionless form is given by :
Now we calculate
The distribution is platykurtic