A Histogram is a Frequency Distribution Diagram. It is a Graph tA Histogram is a Frequency Distribution Diagram. It is a Graph that hat displays the Distribution of Data. It is constructed from Data cdisplays the Distribution of Data. It is constructed from Data collected in a ollected in a Frequency Table, which is ,a Chart that divides the Range of DatFrequency Table, which is ,a Chart that divides the Range of Data into a into several Equal Sections to compare the Frequency of occurrence inseveral Equal Sections to compare the Frequency of occurrence in each each section. The Histogram drawn from the Frequency Table resembles section. The Histogram drawn from the Frequency Table resembles a Bar a Bar Graph, composed of Columns representing the Frequency at which DGraph, composed of Columns representing the Frequency at which Data ata appears in various sections of the range.appears in various sections of the range.

How to make a HistogramHow to make a Histogram

Step Step 1: Gathering Data1: Gathering Data

Step Step 2: Finding the Maximum and Minimum Values 2: Finding the Maximum and Minimum Values

Step Step 3: Determining the Width of the Section (h)3: Determining the Width of the Section (h)

Width of the Section (h) = Width of the Section (h) = Maximum Value Maximum Value -- Minimum ValueMinimum Valuekk

1010 ~~ 2020<< ~~ 250250

77 ~~1212100100 ~~ 250250

66 ~~ 10105050 ~~ 100100

Number ofNumber ofData SectionData Section

kk

Number of Number of Data pointData point

NN

Step 4: Visualizing the Histogram and Determining the Section Boundary Values

The starting point of the Section Boundary Value = (Minimum value) - (Unit). 2

How to determine the Unit:Look at the decimal point of the data carefully.

If there is zero digit after the decimal point: Unit = 1

If there is one digit after the decimal point: Unit = 0.1

If there are two digits after the decimal point: Unit = 0.01

If there are three digits after the decimal point: Unit = 0.001

Go on adding the width of the section (h) to the starting point to find section boundary values.

Step Step 5: 5: Determining the Median Values of the Sections Determining the Median Values of the Sections

Median Value = Median Value = Sum of the two Section BoundariesSum of the two Section Boundaries22

Step Step 6:6: Making a Frequency TableMaking a Frequency Table

Step Step 7: 7: Making a HistogramMaking a Histogram

Make a "Histogram" using the following data:

30.030.029.829.829.229.230.430.429.829.829.329.329.929.930.130.130.030.030.330.330.030.029.829.829.929.929.829.830.630.629629629.829.830.230.229.429.430.330.330.230.229.129.129.729.729.929.930.530.530.030.029.729.730.430.429.529.529.929.930.430.430.630.629.729.729.929.930.030.030.130.130.230.230.330.330.130.129.929.9

29.929.930.130.129.829.830.230.229.929.929.529.530.830.830.230.229.729.730.430.429.529.530.630.630.130.129.929.929.729.730.530.529.929.929.729.729.629.630.030.030.030.029.429.430.430.430.030.029.529.530.030.030.330.329.929.930.030.030.330.329.429.430.430.429.929.930.430.429.829.829.829.830.530.529.629.630.030.029.929.9

Note:- Specified Value is,. 30.2 ± 0.9 mm)

Step 1. Make a Frequency Table with f, u, uf, u2f columns on its right side.

24224266--8080TotalTotal1616444411II30.7530.7530.65 30.65 -- 30.8530.8599545418183366II30.5530.5530.45 30.45 -- 30.6530.658848482424221212IIII30.3530.3530.25 30.25 -- 30.4530.45771010101011101030.1530.1530.05 30.05 -- 30.2530.25660000002323IIIIII29.9529.9529.85 29.85 -- 30.0530.05551414--1414--111414IIIIIIII29.7529.7529.65 29.65 -- 29.8529.85443232--1616--2288IIIIII29.5529.5529.45 29.45 -- 29.6529.65333636--1212--3344IIIIIIII29.3529.3529.25 29.25 -- 29.4529.45223232--88--4422IIII29.1529.1529.05 29.05 -- 29.2529.2511

u2fu2fufufuuffCheckCheckMedianMedianValueValue

Section BoundarySection BoundaryValuesValuesNo.No.

Z

ZZ Z

Z

Process capability for a stable manufacturing process, is the CaProcess capability for a stable manufacturing process, is the Capability of pability of the Process to reach a certain level of Quality. For a Stable Prthe Process to reach a certain level of Quality. For a Stable Process in ocess in which factors affecting the Standard Deviation are properly Contwhich factors affecting the Standard Deviation are properly Controlled, rolled, Process Capability, as measured by the Quality Characteristics oProcess Capability, as measured by the Quality Characteristics of the f the Products of the Process, is usually expressed as the Mean Value Products of the Process, is usually expressed as the Mean Value plus or plus or minus three times the Standard Deviation (x minus three times the Standard Deviation (x + + 3s). If you can freely set a 3s). If you can freely set a mean value (such as standard temperature setting), then you can mean value (such as standard temperature setting), then you can express express process capability as six times the standard deviation from thatprocess capability as six times the standard deviation from that mean mean value.value.Process capability can be shown graphically using Histograms andProcess capability can be shown graphically using Histograms andProcess Capability charts.Process Capability charts.Process Capability Indices numerically express the relation betwProcess Capability Indices numerically express the relation between the een the Distribution and the specification limits.Distribution and the specification limits.Process Capability studies determine whether a process is UnstabProcess Capability studies determine whether a process is Unstable, le, Investigate any sources of instability, determines their causes,Investigate any sources of instability, determines their causes, and take and take action to eliminate such sources of instability.action to eliminate such sources of instability.

Process Capability Inadequate, Improvement shouldbe made.Inadequate1.0> Cp

Process Capability does not Completely meet theSpecification, Process, Control should be Continued.Acceptable1.33 > Cp > 1.0

Process Capability Completely meets SpecificationGoodCp>I.33

AssessmentI EvaluationCp

After all sources of instability have been eliminated from a process, the natural behavior of a, process is called its Process Capability. A process must have an established Process Capability, before it can be improved. ~ Consequently, a Process Capability study must be successfully completed before a process improvement study can have any chance for success. Process .Improvement studies follow the Deming Cycle of Plan, Do, Check and Act. Following table interprets the -valuation for different ranges of the process capability index.

Keep the following points in mind as you codetermine if the Histogram falls well- within the Upper and Lower Limits of the Specifications:

1. Is the Centre of Distribution exactly Half Way between the Upper and Lower Limits of Specifications?

2. Is the Spread of Distribution too large or too small?3. Does any Data appear beyond the Upper Limit or the Lower Limit of

the Distribution?4. Is there ample space for the Distribution within the Specified Limits?

Determining the Distribution Pattern from the shape of HistogramDetermining the Distribution Pattern from the shape of Histogram..

1) Normal Histograms1) Normal HistogramsData obtained from a Stable Process usually produces a HistogramData obtained from a Stable Process usually produces a Histogramthat is Highest in the Centre and Tapers off Symmetrically to ththat is Highest in the Centre and Tapers off Symmetrically to the Left e Left and Right.and Right.

2) Double Peak Histograms2) Double Peak HistogramsDouble Peak appears if you mix Data from different Materials thaDouble Peak appears if you mix Data from different Materials that t have different Averages.have different Averages.

3) Isolated Island Histogram3) Isolated Island HistogramA small Isolated Island appears besides a Normal Histogram. A small Isolated Island appears besides a Normal Histogram. Accidental mixing of Data from another distribution produces thiAccidental mixing of Data from another distribution produces this s type of Histogram.type of Histogram.

4) Cliff Histogram4) Cliff Histogram

The Slope of Histogram ends abruptly in a Steep Cliff. EliminatiThe Slope of Histogram ends abruptly in a Steep Cliff. Eliminating all ng all items that do not meet Specifications is one way to produce thisitems that do not meet Specifications is one way to produce this type type of Histogram.of Histogram.

5) Cogwheel Histograms5) Cogwheel Histograms

Alternating Peaks and Depressions in Adjacent Sections produce aAlternating Peaks and Depressions in Adjacent Sections produce apattern resembling a Cogwheel. Section Widths that are Integral pattern resembling a Cogwheel. Section Widths that are Integral Multiples of the Unit of Measurement of the Scale produce this Multiples of the Unit of Measurement of the Scale produce this shape as a Artifact of the way Scale is read during Measurement.shape as a Artifact of the way Scale is read during Measurement.

Problems On HistogramsProblems On Histograms1. 1. An organization that fills bottles of shampoo ties to remain a sAn organization that fills bottles of shampoo ties to remain a specific pecific

weight of the product. The table gives the weight of 110 bottlesweight of the product. The table gives the weight of 110 bottles that were that were checked at random intervals. Make a tally of these weights and cchecked at random intervals. Make a tally of these weights and construct onstruct a frequency Histogram. (Weight is in Kilograms).a frequency Histogram. (Weight is in Kilograms).

5.976.026.025.986.025.996.035.995.985.996.00

5.986.016.006.015.965.996.016.006.005.995.96

6.025.956.025.996.026.005.995.986.015.996.02

5.996.025.996.025.995.976.016.026.046.015.97

6.006.016.025.986.006.025.595.946.005.986.00

6.006.005.976.005.995.986.006.006.055.986.00

5.995.986,015.986.006.025.995.99.6.016.036.01

5.986.015.995.995.955.976.005.966.006.015.97

5.986.015.975.995.986.015.996.035.995.995.98

5.965.985.996.015.985.995.976.016.015.956.00

1.921.921.901.901.8525

2.001.991.971.941.9224

2.002.001.971.961.9523

2.122.061.911.801.7622

2.001.911.911.891.8721

2.052.00. 1.961.781.7220

2.152.011.981.891.8719

1.991.971.941.901.8118

2.102.001.951.951.9017

2.062.062.011.991.8216

2.031.971.951.951.9015

2.082.001.981.901.8714

2.041.951.921.831.7513

1.991.911.891.821.8012

2.282.062.001.971.90II

2.101.911.931.911.79to

2.052.011.951.911.909

2.051.971.941.921.918

2.102.032.002.001.877

2.001.971.951.881.766

2.142.052.031.951.935

1.931.901.871.831.774

2.071.971.951.871.803

2.011.831.811.811.762

2.202.051.951.931.90I

Duration of Operation Time (Min.)DayProblem 2.

Listed next are 125 readings obtained in a hospital by a motion and time study analyst who took five readings each day for 25 days. Construct a tally sheet. Prepare a table showing cell midpoints, cell boundaries, and observed frequencies plot a frequency histogram.

3. Determine the average median, mode, range, and standard deviation for each group go numbers

(a) 50, 45, 55, 55, 45, 50, 55, 45, 55

(b) 89, 87, 88, 83, 86, 82, 84

(c) 11, 17, 14, 12, 12, 14, 14, 15, 17, 17

(d) 16, 25, 18, 17, 16, 21, 14

(e) 45, 39,42,42,43