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

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Picturisation of Data

N=50Bar Weight

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Fre

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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


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-4Find range of the data

Range of data = Largest value - smallest value

In our caseRange 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 = NNumber of cells, K = N

Step-6Find 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.


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 0.01and TCW = 0.0271428

If multiple factor, M is 1 then we haveM LC = 1 x 0.01 = 0.01This multiple is not nearer to TCW

If multiple factor is 3 then we haveM x LC = 3 x 0.01 = 0.03 This multiple is nearer to TCW

Hence class width, CW = 0.03

Class Boundaries

Step - 7Determine 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

Lowercell boundaryof cell no. 1

Uppercell boundaryof cell no. 1





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) + CWLCB(2) = UCB(1)UCB(2) = LCB(2) + CWLCB(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

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Drawing Histogram

Label vertical axis from zero to a multiple of 1, 2 or 5 to accommodate the largest frequencyLabel vertical axis from zero to a multiple of 1, 2 or 5 to accommodate the largest frequency

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3.37 3.40 3.43 3.46 3.49 3.52 3.55mm

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Label horizontal axis with mid values of the cells, and indicate the dimension of quality characteristicsLabel 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

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3.37 3.40 3.43 3.46 3.49 3.52 3.55

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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

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Drawing HistogramMetal Thickness

N=25

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3.37 3.40 3.43 3.46 3.49 3.52 3.55mm

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Assessing process capability

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Design Tolerance VS Process SpreadF

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47 48 49 50 51 52 53 54kg

Process SpreadProcess Spread

Design ToleranceLSL 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.


47 48 49 50 51 52 53 54

kg

LSL USL

Process is not capable


47 48 49 50 51 52 53 54kg

LSLUSL

Process is just capable


47 48 49 50 51 52 53 54kg

LSL USL

46 55

Process is capable


47 48 49 50 51 52 53 54kg

LSL USL

At the moment process is not capable

Histogram

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