Histogram

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.

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

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.

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.

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.

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.

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.

Data Table - Weight of Bars in kg.

476 513 480 486 508 502 542 489 490 500

507 469 514 537 500 500 479 523 491 500

509 520 474 498 500 478 524 483 503 502

516 489 496 500 487 520 497 490 492 513

500 504 526 502 508 501 528 503 510 512

0

2

4

6

12

14

16

Picturisation of Data

N=50

Bar Weight

8

10

Fre

qu

en

cy

470 480 490 500 510 520 530 540

kg

Key Concept of Histogram

Data always have variation

Variation have pattern

Patterns can be seen easily when summarized

pictorially

Presentation of Distribution

Histogram is represented by a curve. The curve is

known ‘Frequency Distribution’

Study of Histogram

Location of mean of the process

Spread of the process

Shape of the process

While studying histogram look for its

Location of the Process

Process A

Location of

process A

Process B

Location of

Process B

Quality Characteristics

Spread of the Process

Spread of process B

Spread of Process A

Process A

Process B

Shape of the Process

Normal

Distribution Skewed

Distribution

Quality Characteristics

Constructing Histogram

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

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.

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

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’

Addition of Column ‘S’ & Column ‘L’

1 2 3 4 5 S L

3.56 3.46 3.48 3.50 3.42 3.42 3.56

3.43 3.53 3.49 3.44 3.50 3.43 3.53

3.48 3.56 3.50 3.52 3.47 3.47 3.56

3.48 3.46 3.50 3.56 3.38 3.38 3.56

3.41 3.37 3.49 3.45 3.44 3.37 3.49

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

Scanning of Columns ‘S’ & ‘L’

1 2 3 4 5 S L

3.56 3.46 3.48 3.50 3.42 3.42 3.56

3.43 3.53 3.49 3.44 3.50 3.43 3.53

3.48 3.56 3.50 3.52 3.47 3.47 3.56

3.48 3.46 3.50 3.56 3.38 3.38 3.56

3.41 3.37 3.49 3.45 3.44 3.37 3.49

Overall largest reading = 3.56

Overall smallest reading = 3.37

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

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

Table for Choosing Number of Cells

Number of observation(N)

Number of cells( K )

Under 50 5 to 7

50 - 100 6 to 10

101 - 250 7 to 12

More than 250 10 to 20

Method No. 1

Alternative Methods for Deciding No. of Cells

Method No. 2

Method No 3

Number of cells, K = 1 + 2.33 Log 10

N

Number of cells, K = N

Step-6

Find temporary cell width, TCW

Temporary Cell Width

TCW = Range (R)

Number of cells chosen (K)

= 0.19

7

= 0.0271423

Rounding of Temporary Cell Width

For ease of plotting

For getting distinct cell boundary

Temporary cell width, TCW needs rounding off.

Construction of Histogram

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

Step - 6

Round off TCW to get class width

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

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

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

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.

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.

Elements of Histogram

Cell

No. 2

Cell

No. 1

Cell

No. 3

CW CW CW

Lower

cell boundary

of cell no. 1

Upper

cell boundary

of cell no. 1

Lower

cell boundary

of cell no. 2

Upper

cell boundary

of cell no. 2

Upper

cell boundary

of cell no. 3

Lower

cell boundary

of cell no. 3

Continuous Scale

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

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

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.

SN Cell BoundaryMid

ValueTally

MarksFrequency

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

Filling of Frequency Column

Count the number of tally marks in each cell and

enter the count in ‘Frequency’ column

Drawing Histogram

Draw horizontal axis

Draw vertical axis

8

6

Drawing Histogram

Label vertical axis from zero to a multiple of 1, 2 or

5 to accommodate the largest frequency

1

2

4

0

3

7

5

Fre

que

ncy

3.37 3.40 3.43 3.46 3.49 3.52 3.55

mm

9

Label horizontal axis with mid values of the cells,

and indicate the dimension of quality characteristics

Drawing Histogram

Leave one cell

width space from

vertical axis

1

2

4

0

3

7

5

6

8

Fre

que

ncy

3.37 3.40 3.43 3.46 3.49 3.52 3.55

mm

9

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

1

2

4

0

3

7

5

Drawing Histogram

Metal Thickness

N=25

6

8

Fre

que

ncy

3.37 3.40 3.43 3.46 3.49 3.52 3.55

mm

9

Assessing process capability

8

0

2

4

6

10

12

14

16

Design Tolerance VS Process Spread F

requ

en

cy

47 48 49 50 51 52 53 54 kg

Process Spread

Design Tolerance LSL USL

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.

Assessing Process Capability

47 48 49 50 51 52 53 54

kg

LSL USL

Process is not capable

Assessing Process Capability

47 48 49 50 51 52 53 54 kg

LSL USL

Process is just capable

Assessing Process Capability

47 48 49 50 51 52 53 54 kg

LSL USL

46 55

Process is capable

Assessing Process Capability

47 48 49 50 51 52 53 54

kg

LSL USL

At the moment process is not capable