2. What is a Histogram?
Histogram is a visual tool for presenting variable data . It
organises data to describe the process performance.
Additionally histogram shows the amount and pattern of the
variation from the process.
Histogram offers a snapshot in time of the process performance.
3. Why do We Get Variation?
Variation is essentially law of nature.
Output quality characteristics depends upon the input parameters.
It is impossible to keep input parameters constant. There will be
always variation in the input parameters. Since there is variation in
the input parameters, there is also variation in the output
characteristics
4. Law of Nature
In nature there is always variation. Take case measurement of the
following:
height of adult male in a city.
weight of 15 years old boy in a town.
weight of bars 5 meter long 25 mm dia.
volume in 300 cc soft drink bottle.
number of minutes required to fill an invoice.
5. Case when Data Does Not Show Variation
There could be two reasons when data do not show variation:
a) Measuring devices are insensitive to spot variation.
b) Too much rounding off the data while recording.
6. Insensitive Measuring Device
If the measuring device is not sensitive, enough to respond to
small changes in value of the quality characteristics, variation will
not be reflected in the data. For example:
Weighing gold chains by using weighing scale used for
vegetables.
7. Too Much Rounding Off During Recording
It could also be possible that too much rounding off might
have been carried while recording the measurements.
This normally happens when the column in data recording
sheet is not wide enough to record all the decimal places of
measurements. Because of paucity of the space, workmen
round off observations on their own.
8. Definition of Histogram
A histogram is a graphical summary of variation in a
set of data.
The pictorial nature of the histogram enables us to see
patterns that are difficult to see in a table of numbers.
18. Basic Elements for Construction of Histogram
For constructing the histogram we need to know the
following:
Lowest value of the data set
Highest value of the data set
Approximate number of cells histogram have
Cell width
Lower cell boundary of first cell
19. Finding Lowest & Largest Value in Data Set
If the number of observations in the data set is small, then finding
smallest and largest value is not a problem.
However, if the number of observations is large, then we require
an easier way to get smallest value and largest value in the data
set. This can be achieved by grouping the data in rows, columns
and then scanning.
20. Organizing Data in Rows & Columns
Step - 1
Organise the data in a group of 5 or 10
1 2 3 4 5
3.56 3.46 3.48 3.42 3.43
3.48 3.56 3.50 3.52 3.47
3.48 3.46 3.50 3.56 3.38
3.41 3.37 3.49 3.45 3.44
3.50 3.49 3.46 3.46 3.42
21. Construction of Histogram
Step - 2
Generate 2 more columns to record
Smallest value in each row in column ‘S’
Largest value in each row in column ‘L’
23. Construction of Histogram
Step-3
Scan column ‘S’ to find smallest value in that column, S. S is
overall smallest value in the data set.
Scan column ‘L’ to find largest value in that column, L. L is overall
largest value in the data set
25. Range of the Data Set
Step-4
Find range of the data
Range of data = Largest value - smallest value
In our case
Range R = L - S
= 3.56 - 3.37
= 0.19
26. Initial Number of Cells in Histogram
Step-5
Decide the initial number of cells, say K, a histogram shall
have.
Number of cells a histogram can have, depends upon the number of
observations N, histogram is representing. There are three methods
to decide initial number of cells.
Note: The number of cells, K initially chosen may change when
histogram is finally made
27. Table for Choosing Number of Cells
Method No. 1
Number of observation Number of cells
(N) (K)
Under 50 5 to 7
50 - 100 6 to 10
101 - 250 7 to 12
More than 250 10 to 20
28. Alternative Methods for Deciding No. of Cells
Method No. 2
Number of cells, K = 1 + 2.33 Log 10 N
Method No 3
Number of cells, K = N
30. Rounding of Temporary Cell Width
Temporary cell width, TCW needs rounding off.
For ease of plotting
For getting distinct cell boundary
31. Construction of Histogram
Step - 6
Round off TCW to get class width
Rounding off of TCW, should be in the multiple of 1 or 3 or 5 of
least count.
The multiple should be nearer to TCW
32. Least Count of the Data
1 2 3 4 5
3.56 3.46 3.48 3.42 3.43
3.48 3.56 3.50 3.52 3.47
3.48 3.46 3.50 3.56 3.38
3.41 3.37 3.49 3.45 3.44
3.50 3.49 3.46 3.46 3.42
Least count of the data is 0.01
33. Procedure for Getting Class Width
In our case least count of the data,
LC is 0.01
and TCW = 0.0271428
If multiple factor, M is 1 then we have
M × LC = 1 x 0.01 = 0.01
This multiple is not nearer to TCW
If multiple factor is 3 then we have
M x LC = 3 x 0.01 = 0.03
This multiple is nearer to TCW
Hence class width, CW = 0.03
34. Class Boundaries
Step - 7
Determine class boundaries
Class boundaries are necessary for making tally sheet.
Frequency obtained in tally sheet is utilised for making histogram.
Class boundaries should be distinct
35. Distinct Class Boundaries
Distinct class boundaries are the one, on which no individual data
lies.
With the distinct class boundary the data will enter in a particular
cell only.
36. Nomenclature of Cell Boundaries
Let LCB(1), LCB(2), … are the lower cell boundaries of cell no.1,
cell No. 2…. respectively.
Let UCB(1), UCB(2), … are the upper cell boundaries of cell no.1,
cell No. 2…. respectively.
37. Elements of Histogram
Upper
Lower
cell boundary
cell boundary
of cell no. 2
of cell no. 2
Upper Lower
cell boundary cell boundary
of cell no. 1 of cell no. 3
Cell
No. 2
Lower Cell Upper
cell boundary Cell No. 3 cell boundary
of cell no. 1 No. 1 of cell no. 3
CW CW CW
Continuous Scale
38. Calculation of Cell Boundaries
If we know the lower cell boundary of cell No.1, LCB(1), and class
width, CW we can find other cell boundaries as follows:
UCB(1) = LCB(1) + CW
LCB(2) = UCB(1)
UCB(2) = LCB(2) + CW
LCB(3) = UCB(2)
and so on
39. Getting Lower Cell Boundary of Cell No.1
Choose a starting value A, which is slightly lower or equal to
smallest value, S. Value of S in our case is 3.37
We can take A = 3.37
LCB = A - ( CW / 2 )
= 3.37 - ( 0.03 / 2 )
= 3.355
40. Getting Cell Boundaries
UCB(1) = LCB(1) + CW
= 3.355 + 0.03 = 3.385
LCB(2) = UCB (1) = 3.385
UCB(2) = LCB(2) + CW
= 3.385 + 0.03 = 3.415
Continue finding cell boundaries, till a particular upper cell boundary
is greater than the largest value of data set.
41. Filling of Frequency Column
Count the number of tally marks in each cell and
enter the count in ‘Frequency’ column
Mid Tally
SN Cell Boundary Frequency
Value Marks
1 3.355 - 3.385 3.37 2
2 3.385 - 3.415 3.40 2
3 3.415 - 3.444 3.43 3
4 3.445 - 3.475 3.46 4
5 3.475 - 3.505 3.49 8
6 3.505 – 3.535 3.52 4
7 3.535 - 3.565 3.55 2
43. Drawing Histogram
9
8 Label vertical axis from zero to a multiple of 1, 2
or 5 to accommodate the largest frequency
7
Frequency
6
5
4
3 Label horizontal axis with mid values of the cells,
and indicate the dimension of quality characteristics
2
1
0
3.37 3.40 3.43 3.46 3.49 3.52 3.55
mm
44. Drawing Histogram
9
8
7
Frequency
6
5
Leave one cell
4 width space from
3 vertical axis
2
1
0
3.37 3.40 3.43 3.46 3.49 3.52 3.55
mm
45. Drawing Histogram
Draw bars to represent frequency in each cell. Height of bars is
equal to number of data in each cell.
Title the chart.
Indicate total number of observations
46. Drawing Histogram
9
Metal Thickness
8
N=25
7
Frequency
6
5
4
3
2
1
0
3.37 3.40 3.43 3.46 3.49 3.52 3.55
mm
48. Design Tolerance VS Process Spread
LSL USL
16
Design Tolerance
14 Process Spread
Frequency
12
10
8
6
4
2
0
47 48 49 50 51 52 53 54
kg
49. Assessing Process Capability
Process capability is a comparison between design tolerance and
spread of the process.
Whenever design tolerance is more than process spread, then the
process is capable.
Whenever design tolerance is less than the spread of the process,
then the process is not capable.
