Module 11 - Histogram

Basic Tools for Process Improvement

HISTOGRAM 1

Module 11

HISTOGRAM


2 HISTOGRAM

What is a Histogram?

A Histogram is a vertical bar chart that depicts the distribution of a set of data. UnlikeRun Charts or Control Charts, which are discussed in other modules, a Histogramdoes not reflect process performance over time. It's helpful to think of a Histogramas being like a snapshot, while a Run Chart or Control Chart is more like a movie(Viewgraph 1).

When should we use a Histogram?

When you are unsure what to do with a large set of measurements presented in atable, you can use a Histogram to organize and display the data in a more user-friendly format. A Histogram will make it easy to see where the majority of valuesfalls in a measurement scale, and how much variation there is. It is helpful toconstruct a Histogram when you want to do the following (Viewgraph 2):

! Summarize large data sets graphically. When you look at Viewgraph 6,you can see that a set of data presented in a table isn’t easy to use. You canmake it much easier to understand by summarizing it on a tally sheet(Viewgraph 7) and organizing it into a Histogram (Viewgraph 12).

! Compare process results with specification limits. If you add theprocess specification limits to your Histogram, you can determine quicklywhether the current process was able to produce "good" products. Specification limits may take the form of length, weight, density, quantity ofmaterials to be delivered, or whatever is important for the product of a givenprocess. Viewgraph 14 shows a Histogram on which the specification limits,or "goalposts," have been superimposed. We’ll look more closely at theimplications of specification limits when we discuss Histogram interpretationlater in this module.

! Communicate information graphically. The team members can easilysee the values which occur most frequently. When you use a Histogram tosummarize large data sets, or to compare measurements to specificationlimits, you are employing a powerful tool for communicating information.

! Use a tool to assist in decision making. As you will see as we movealong through this module, certain shapes, sizes, and the spread of data havemeanings that can help you in investigating problems and making decisions. But always bear in mind that if the data you have in hand aren’t recent, or youdon’t know how the data were collected, it’s a waste of time trying to chartthem. Measurements cannot be used for making decisions or predictionswhen they were produced by a process that is different from the current one,or were collected under unknown conditions.

HISTOGRAM VIEWGRAPH 1

What Is a Histogram?

• A bar graph that shows the distribution of data• A snapshot of data taken from a process

0 5 10 15 20 25 30 35 40 45 50 55 60

0

20

40

60

80

100


When Are Histograms Used?

• Summarize large data sets graphically

• Compare measurements to specifications

• Communicate information to the team

• Assist in decision making


HISTOGRAM 3


4 HISTOGRAM

What are the parts of a Histogram?

As you can see in Viewgraph 3, a Histogram is made up of five parts:

1. Title: The title briefly describes the information that is contained in theHistogram.

2. Horizontal or X-Axis: The horizontal or X-axis shows you the scale ofvalues into which the measurements fit. These measurements are generallygrouped into intervals to help you summarize large data sets. Individual datapoints are not displayed.

3. Bars: The bars have two important characteristics—height and width. Theheight represents the number of times the values within an interval occurred. The width represents the length of the interval covered by the bar. It is thesame for all bars.

4. Vertical or Y-Axis: The vertical or Y-axis is the scale that shows you thenumber of times the values within an interval occurred. The number of timesis also referred to as "frequency."

5. Legend: The legend provides additional information that documents wherethe data came from and how the measurements were gathered.


1 Title 2 Horizontal / X-axis3 Bars 4 Vertical / Y-axis5 Legend

0 5 10 15 20 25 30 35 40 45 50 55 60

DAYS OF OPERATION PRIOR TOFAILURE FOR AN HF RECEIVER

DAYS OF OPERATION

MEAN TIME BETWEEN FAILURE (IN DAYS) FOR R-1051 HF RECEIVERData taken at SIMA, Pearl Harbor, 15 May - 15 July 94

Parts of a Histogram

1

3

2

FREQUENCY

4

5

0

20

40

60

80

100


HISTOGRAM 5


6 HISTOGRAM

How is a Histogram constructed?

There are many different ways to organize data and build Histograms. You cansafely use any of them as long as you follow the basic rules. In this module, we willuse the nine-step approach (Viewgraphs 4 and 5) described on the following pages.

EXAMPLE: The following scenario will be used as an example to provide data aswe go through the process of building a Histogram step by step:

During sea trials, a ship conducted test firings of its MK 75, 76mm gun. The ship fired 135 rounds at a target. An airborne spotter provided accurate rake data to assess the fall of shot both long and short of the target. The ship computed what constituted a hit for the test firing as:

From 60 yards short of the target

To 300 yards beyond the target


Step 1 - Count number of data points

Step 2 - Summarize on a tally sheet

Step 3 - Compute the range

Step 4 - Determine number of intervals

Step 5 - Compute interval width

Constructing a Histogram


Constructing a Histogram

Step 6 - Determine interval starting points

Step 7 - Count number of points in each interval

Step 8 - Plot the data

Step 9 - Add title and legend


HISTOGRAM 7


8 HISTOGRAM

Step 1 - Count the total number of data points you have listed. Suppose your team collected data on the miss distance for the gunnery exercise described inthe example. The data you collected was for the fall of shot both long and short ofthe target. The data are displayed in Viewgraph 6. Simply counting the totalnumber of entries in the data set completes this step. In this example, there are135 data points.

Step 2 - Summarize your data on a tally sheet. You need to summarize yourdata to make it easy to interpret. You can do this by constructing a tally sheet.

First, identify all the different values found in Viewgraph 6 (-160, -010. . .030,220, etc.). Organize these values from smallest to largest (-180, -120. . .380,410).

Then, make a tally mark next to the value every time that value is present inthe data set.

Alternatively, simply count the number of times each value is present in thedata set and enter that number next to the value, as shown in Viewgraph 7.

This tally helped us organize 135 mixed numbers into a ranked sequence of 51values. Moreover, we can see very easily the number of times that each valueappeared in the data set. This data can be summarized even further by formingintervals of values.


Step 1 - Count the total number of data points

-180 30 190 380 330 140 160 270 10 - 90- 10 30 60 230 90 120 10 50 250 180-130 220 170 130 - 50 - 80 180 100 110 200 260 190 -100 150 210 140 -130 130 150 370 160 180 240 260 - 20 - 80 30 80 240 130 210 40 70 - 70 250 360 120 - 60 - 30 200 50 20 30 280 410 70 - 10 20 130 170 140 220 - 40 290 90 100 - 30 340 20 80 210 130 350 250 - 20 230 180 130 - 30 210 -30 80 270 320 30 240 120 100 20 70 300 260 20 40 - 20 250 310 40 200 190 110 -30 50 240 180 50 130 200 280 60 260 70 100 140 80 190 100 270 140 80 110 130 120 30 70

TOTAL = 135

Number of yards long (+ data) and yards short (- data) that a gun crew missed its target.

How to Construct a Histogram


Step 2 - Summarize the data on a tally sheet


DATA TALLY DATA TALLY DATA TALLY DATA TALLY DATA TALLY

- 180 3 90

- 130 2 100

- 100 2 110

- 90 5 120

- 80 6 130

- 70 3 140

- 60 4 150 1

- 50 2 160

- 40 5 170

- 30

1

2

1

1

2

1

1

1

1

5

- 20

- 10

10

20

30

40

50

60

70

80 5 180

2

5

34

8

5

2

2

2

5

190

200

210

220

230

240

250

260

270

280

4

4

4

2

2

4

4

4

3

2

290

300

310

320

330

340

350

360

370

380

410

1

1

1

1

1

1

1

1

1

1


HISTOGRAM 9


10 HISTOGRAM

Step 3 - Compute the range for the data set. Compute the range by subtractingthe smallest value in the data set from the largest value. The range representsthe extent of the measurement scale covered by the data; it is always a positivenumber. The range for the data in Viewgraph 8 is 590 yards. This number isobtained by subtracting -180 from +410. The mathematical operation brokendown in Viewgraph 8 is:

+410 - (-180) = 410 + 180 = 590

Remember that when you subtract a negative (-) number from another number itbecomes a positive number.

Step 4 - Determine the number of intervals required. The number of intervalsinfluences the pattern, shape, or spread of your Histogram. Use the followingtable (Viewgraph 9) to determine how many intervals (or bars on the bar graph)you should use.

If you have this Use this numbermany data points: of intervals:

Less than 50 5 to 7

50 to 99 6 to 10

100 to 250 7 to 12

More than 250 10 to 20

For this example, 10 has been chosen as an appropriate number of intervals.


Largest value = + 410 yards past target

Smallest value = - 180 yards short of target

Range of values = 590 yards

Step 3 - Compute the range for the data set


Calculation: + 410 - (- 180) = 410 + 180 = 590


IF YOU HAVE THISMANY DATA POINTS

USE THIS NUMBEROF INTERVALS:

Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals

Step 4 - Determine the number of intervalsrequired



HISTOGRAM 11


12 HISTOGRAM

Step 5 - Compute the interval width. To compute the interval width (Viewgraph 10), divide the range (590) by the number of intervals (10). When computing theinterval width, you should round the data up to the next higher whole number tocome up with values that are convenient to use. For example, if the range of datais 17, and you have decided to use 9 intervals, then your interval width is 1.88. You can round this up to 2.

In this example, you divide 590 yards by 10 intervals, which gives an intervalwidth of 59. This means that the length of every interval is going to be 59 yards. To facilitate later calculations, it is best to round off the value representing thewidth of the intervals. In this case, we will use 60, rather than 59, as the intervalwidth.

Step 6 - Determine the starting point for each interval. Use the smallest datapoint in your measurements as the starting point of the first interval. The startingpoint for the second interval is the sum of the smallest data point and the intervalwidth. For example, if the smallest data point is -180, and the interval width is 60,the starting point for the second interval is -120. Follow this procedure(Viewgraph 11) to determine all of the starting points (-180 + 60 = -120; -120 + 60= -60; etc.).

Step 7 - Count the number of points that fall within each interval. These arethe data points that are equal to or greater than the starting value and less thanthe ending value (also illustrated in Viewgraph 11). For example, if the firstinterval begins with -180 and ends with -120, all data points that are equal to orgreater than -180, but still less than -120, will be counted in the first interval. Keepin mind that EACH DATA POINT can appear in only one interval.


59IntervalWidth

Range

Number ofIntervals

=590

10=

Use 10 for thenumber of intervals

Use 10 for thenumber of intervals

Round upto 60

=

Step 5 - Compute the interval width



INTERVALNUMBER

STARTINGVALUE

ENDINGVALUE

1

2

3

4

5

6

7

8

9

10

-180

-120

-060

000

060

120

180

240

300

360

INTERVALWIDTH

60

60

60

60

60

60

60

60

60

60

-120

-060

000

060

120

180

240

300

360

420

NUMBER OFCOUNTS

3

5

13

20

22

24

20

18

6

4

Equal to or greater than theSTARTING VALUE

But less than theENDING VALUE

Step 6 - Determine the starting point of each intervalStep 7 - Count the number of points in each interval



HISTOGRAM 13


14 HISTOGRAM

Step 8 - Plot the data. A more precise and refined picture comes into view once you plot your data (Viewgraph 12). You bring all of the previous steps togetherwhen you construct the graph.

! The horizontal scale across the bottom of the graph contains the intervals thatwere calculated previously.

! The vertical scale contains the count or frequency of observations within eachof the intervals.

! A bar is drawn for the height of each interval. The bars look like columns.

! The height is determined by the number of observations or percentage of thetotal observations for each of the intervals.

! The Histogram may not be perfectly symmetrical. Variations will occur. Askyourself whether the picture is reasonable and logical, but be careful not to letyour preconceived ideas influence your decisions unfairly.

Step 9 - Add the title and legend. A title and a legend provide the who, what,when, where, and why (also illustrated in Viewgraph 12) that are important forunderstanding and interpreting the data. This additional information documentsthe nature of the data, where it came from, and when it was collected. The legendmay include such things as the sample size, the dates and times involved, whocollected the data, and identifiable equipment or work groups. It is important toinclude any information that helps clarify what the data describes.


USS CROMMELIN (FFG-37), PACIFIC MISSILE FIRING RANGE, 135 BL&P ROUNDS/MOUNT 31, 25 JUNE 94LEGEND:

Step 8 - Plot the data Step 9 - Add the title and legend


0

5

10

15

20

25

-180 -120 -060 000 060 120 180 240 300 360 420

TARGET

YARDS LONGYARDS SHORT

MISS DISTANCE FOR MK 75 GUN TEST FIRINGHITS

MISSESSHOT

COUNT

MISSES


HISTOGRAM 15


16 HISTOGRAM

How do we interpret a Histogram?

A Histogram provides a visual representation so you can see where most of themeasurements are located and how spread out they are. Your Histogram mightshow any of the following conditions (Viewgraph 13):

! Most of the data were on target, with very little variation from it, as inViewgraph 13A.

! Although some data were on target, many others were dispersed away fromthe target, as in Viewgraph 13B.

! Even when most of the data were close together, they were located off thetarget by a significant amount, as in Viewgraph 13C.

! The data were off target and widely dispersed, as in Viewgraph 13D.

This information helps you see how well the process performed and how consistent itwas. You may be thinking, "So what? How will this help me do my job better?" Well, with the results of the process clearly depicted, we can find the answer to a vitalquestion:

Did the process produce goods and services which are within specification limits?

Looking at the Histogram, you can see, not only whether you were withinspecification limits, but also how close to the target you were (Viewgraph 14).


Target Target

Target Target

Interpreting Histograms Location and Spread of Data

A

DC

B


Interpreting HistogramsIs Process Within Specification

Limits?

TargetLSL USL LSL USLTarget

WITHIN LIMITS OUT OF SPEC

LSL = Lower specification limitUSL = Upper specification limit


HISTOGRAM 17


18 HISTOGRAM

Portraying your data in a Histogram enables you to check rapidly on the number, orthe percentage, of defects produced during the time you collected data. But unlessyou know whether the process was stable (Viewgraph 15), you won’t be able topredict whether future products will be within specification limits or determine acourse of action to ensure that they are.

A Histogram can show you whether or not your process is producing products orservices that are within specification limits. To discover whether the process isstable, and to predict whether it can continue to produce within spec limits, you needto use a Control Chart (see the Control Chart module). Only after you havediscovered whether your process is in or out of control can you determine anappropriate course of action—to eliminate special causes of variation, or to makefundamental changes to your process.

There are times when a Histogram may look unusual to you. It might have more thanone peak, be discontinued, or be skewed, with one tail longer than the other, asshown in Viewgraph 16. In these circumstances, the people involved in the processshould ask themselves whether it really is unusual. The Histogram may not besymmetrical, but you may find out that it should look the way it does. On the otherhand, the shape may show you that something is wrong, that data from severalsources were mixed, for example, or different measurement devices were used, oroperational definitions weren't applied. What is really important here is to avoidjumping to conclusions without properly examining the alternatives.


Interpreting HistogramsProcess Variation

Target

Day 1

Target

Day 2

Target

Day 3 Day 4

Target


Skewed(not symmetrical)

Discontinued

Interpreting HistogramsCommon Histogram Shapes

Symmetrical(mirror imaged)


HISTOGRAM 19


20 HISTOGRAM

How can we practice what we've learned?

Two exercises are provided that will take you through the nine steps for developing aHistogram. On the four pages that follow the scenario for Exercise 1 you will find aset of blank worksheets (Viewgraphs 17 through 23) to use in working through bothof the exercises in this module.

You will find a set of answer keys for Exercise 1 after the blank worksheets, and forExercise 2 after the description of its scenario. These answer keys represent only one possible set of answers. It's all right for you to choose an interval width or anumber of intervals that is different from those used in the answer keys. Eventhough the shape of your Histogram may vary somewhat from the answer key'sshape, it should be reasonably close unless you used a very different number ofintervals.

EXERCISE 1: The source of data for the first exercise is the following scenario. Alist of the data collected follows this description. Use the blank worksheets inViewgraphs 17 through 23 to do this exercise. You will find answer keys inViewgraphs 24 through 30.

Your corpsman is responsible for the semiannual PhysicalReadiness Test (PRT) screening for percent body fat. Priorto one PRT, the corpsman recorded the percent of body fat for the 80 personnel assigned to the command. These arethe data collected:

PERCENT BODY FAT RECORDED

11 22 15 7 13 20 25 12 16 19

4 14 11 16 18 32 10 16 17 10

8 11 23 14 16 10 5 21 26 10

23 12 10 16 17 24 11 20 9 13

24 10 16 18 22 15 13 19 15 24

11 20 15 13 9 18 22 16 18 9

14 20 11 19 10 17 15 12 17 11

17 11 15 11 15 16 12 28 14 13


Step 1 - Count the number of data points

WORKSHEET

TOTAL NUMBER =


WORKSHEET


VALUE TALLY VALUE TALLY VALUE TALLY VALUE TALLY VALUE TALLY


HISTOGRAM 21


Largest value = _______________

Smallest value = _______________

________________________________________

Range of values = _______________


WORKSHEET


Step 4 - Determine the number of intervals

WORKSHEET



Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals


22 HISTOGRAM



WORKSHEET

IntervalWidth

Range

Number ofIntervals

= =

Round up tonext higher

whole number

=


Step 6 - Determine the starting point of each interval

WORKSHEET

INTERVAL STARTING INTERVAL ENDING NUMBERNUMBER VALUE WIDTH VALUE OF COUNTS

123456789

10

Step 7 - Count the number of points in each interval


HISTOGRAM 23


Step 8 - Plot the dataStep 9 - Add title and legend

WORKSHEET


24 HISTOGRAM



EXERCISE 1 ANSWER KEY

11 22 15 7 13 20 25 12 16 19

4 14 11 16 18 32 10 16 17 10

8 11 23 14 16 10 5 21 26 10

23 12 10 16 17 24 11 20 9 13

24 10 16 18 22 15 13 19 15 24

11 20 15 13 9 18 22 16 18 9

14 20 11 19 10 17 15 12 17 11

17 11 15 11 15 16 12 28 14 13

TOTAL = 80




0 0 1 0 2 0 3 0 4 1 5 1 6 0 7 1 8 1 9 310 7

% FAT NO. OF PERS

11 912 413 514 415 716 817 518 419 320 421 1

% FAT NO. OF PERS

22 323 224 325 126 127 028 129 030 031 032 1

% FAT NO. OF PERS


HISTOGRAM 25




Largest value = 32 Percent body fat

Smallest value = 4 Percent body fat

_________________________________________

Range of values = 28 Percent body fat




Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals




26 HISTOGRAM



Equal to or greater thanthe STARTING VALUE

But less thanthe ENDING VALUE

Step 6 - Determine the starting point of each interval



1 4 + 4 8 32 8 + 4 12 203 12 + 4 16 204 16 + 4 20 205 20 + 4 24 106 24 + 4 28 57 28 + 4 32 18 32 + 4 36 1


3.5



IntervalWidth

Range

Number ofIntervals

= =

Round upto 4

=28

8

Use 8 for the numberof intervals


HISTOGRAM 27



LEGEND: USS LEADER (MSO-490), 25 JUNE 94, ALL 80 PERSONNEL SAMPLED

4 8 12 16 20 24 28 32 360

JUNE 94 PRT PERCENT BODY FATSATISFACTORY % BODY FAT

PERCENT BODY FAT

NO

. OF

PE

RS

ON

NE

L

0

4

6

8

10

12

14

16

18

20

2

Step 8 - Plot the data Step 9 - Add title and legend


28 HISTOGRAM


HISTOGRAM 29

EXERCISE 2: The source of data for the second exercise is the following scenario. A listing of the data collected follows this description. Use the blank worksheets inViewgraphs 17 through 23 to do this exercise. You will find answer keys inViewgraphs 31 through 37.

A Marine Corps small arms instructor was performing an analysis of 9 mm pistol marksmanship scores to improvetraining methods. For every class of 25, the instructorrecorded the scores for each student who occupied the first four firing positions at the small arms range. Theinstructor then averaged the scores for each class,maintaining a database on 105 classes. These are thedata collected:

AVERAGE SMALL ARMS SCORES

160 190 155 300 280 185 250 285 200 165

175 190 210 225 275 240 170 185 215 220

270 265 255 235 170 175 185 195 200 260

180 245 270 200 200 220 265 270 250 230

255 180 260 240 245 170 205 260 215 185

255 245 210 225 225 235 230 230 195 225

230 255 235 195 220 210 235 240 200 220

195 235 230 215 225 235 225 200 245 230

220 215 225 250 220 245 195 235 225 230

210 240 215 230 220 225 200 235 215 240

220 230 225 215 225


160 190 155 300 280 185 250 285 200 165

175 190 210 225 275 240 170 185 215 220

270 265 255 235 170 175 185 195 200 260

180 245 270 200 200 220 265 270 250 230

255 180 260 240 245 170 205 260 215 185

255 245 210 225 225 235 230 230 195 225

230 255 235 195 220 210 235 240 200 220

195 235 230 215 225 235 225 200 245 230

220 215 225 250 220 245 195 235 225 230

210 240 215 230 220 225 200 235 215 240

220 230 225 215 225


TOTAL = 105



155 1

160 1

165 1

170 3

175 2

180 2

185 4

190 2

195 5

200 7

205 1

210 4

215 7

220 8

225 11

230 9

235 8

240 5

245 5

250 3

255 4

260 3

265 2

270 3

275 1

280 1

285 1

290 0

295 0

300 1

Step 2 - Summarize the data on a tally sheetEXERCISE 2 ANSWER KEY

SCORE TALLY SCORE TALLY SCORE TALLY


30 HISTOGRAM




Largest value = 300 Points

Smallest value = 155 Points

__________________________________

Range of values = 145 Points




Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals




HISTOGRAM 31


EXERCISE 2 ANSWER KEYStep 6 - Determine the starting point of each interval


Equal to or greater thanthe STARTING VALUE

But less thanthe ENDING VALUE


1 155 + 15 170 32 170 + 15 185 73 185 + 15 200 114 200 + 15 215 125 215 + 15 230 266 230 + 15 245 227 245 + 15 260 128 260 + 15 275 89 275 + 15 290 3

10 290 + 15 300 1


14.5



IntervalWidth

Range

Number ofIntervals

= =

Round upto 15

=145

10

Use 10 for the numberof intervals


32 HISTOGRAM


155 170 185 200 215 230 245 260 275 290 300

05

1015202530

SCORES

NO

. OF

PE

RS

ON

NE

L

MARKSMANSHIP SCORES FOR 9mm PISTOL

LEGEND: MCBH KANEOHE BAY, HI; AVERAGE OF 4 SCORES PER CLASS, 105 CLASSES, 1 JUNE 94 - 15 JULY 94


Step 9 - Add title and legend

Step 8 - Plot the data


HISTOGRAM 33


34 HISTOGRAM

REFERENCES:

1. Brassard, M. (1988). The Memory Jogger, A Pocket Guide of Tools forContinuous Improvement, pp. 36 - 43. Methuen, MA: GOAL/QPC.

2. Department of the Navy (November 1992), Fundamentals of Total QualityLeadership (Instructor Guide), pp. 6-44 - 6-47. San Diego, CA: Navy PersonnelResearch and Development Center.

3. Department of the Navy (September 1993). Systems Approach to ProcessImprovement (Instructor Guide), pp. 10-17 - 10-38. San Diego, CA: OUSN TotalQuality Leadership Office and Navy Personnel Research and DevelopmentCenter.

4. Naval Medical Quality Institute (Undated). Total Quality Leader's Course (StudentGuide), pp. U-26 - U-28. Bethesda, MD.

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Wh

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•A

bar graph that shows the distribution of data

•A

snapshot of data taken from a process

05

1015

2025

3035

4045

5055

60

0

20 40 60 80

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•C

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•A

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Data taken at S

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Parts o

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FREQUENCY45

0 20 40 60 80

100

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Step 8 - P

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-180 30

190 380

330 140

160 270

10- 90

- 10 30

60 230

90 120

10 50

250 180

-130220

170 130

- 50- 80

180 100

110 200

260190

-100 150

210 140

-130 130

150 370

160180

240 260

- 20- 80

30 80

240 130

210 40

70- 70

250 360

120- 60

- 30 200

50 20

30 280

410 70

- 10 20

130 170

140220

- 40 290

90 100

- 30 340

20 80

210130

350 250

- 20 230

180 130

- 30 210

-30 80

270 320

30 240

120 100

20 70

300260

20 40

- 20 250

310 40

200 190

110 -30

50 240

180 50

130 200

280 60

260 70

100 140

80 190

100 270

140 80

110130

120 30

70

TOTA

L = 135

Num

ber of yards long (+ data) and yards short (- data) that a gun crew m

issed its target.

How

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istogram

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How

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TATA

LLY

- 1803

90

- 1302

100

- 1002

110

- 905

120

- 806

130

- 703

140

- 604

1501

- 502

160

- 405

170

- 30

1211211115

- 20

- 1010203040506070805

180

2534852225

190

200

210

220

230

240

250

260

270

280

4442244432

290

300

310

320

330

340

350

360

370

380

410

1111111111

HIS

TOG

RA

MV

IEW

GR

AP

H 8

Largest value= + 410 yards past target

Sm

allest value= - 180 yards short of target

Range of values

= 590 yards

Step

3 - Co

mp

ute th

e rang

e for th

e data set

How

to Construct a H

istogram

Calculation: + 410 - (- 180) = 410 + 180 = 590

HIS

TOG

RA

MV

IEW

GR

AP

H 9

IF Y

OU

HA

VE

TH

ISM

AN

Y D

AT

A P

OIN

TS

US

E T

HIS

NU

MB

ER

OF

INT

ER

VA

LS:

Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals

Step

4 - Determ

ine th

e nu

mb

er of in

tervalsreq

uired

How

to Construct a H

istogram

HIS

TOG

RA

MV

IEW

GR

AP

H 10

59IntervalW

idth

Range

Num

ber ofIntervals

=590

10=

Use 10 for the

number of intervals

Use 10 for the

number of intervals

Round upto 60

=

Step

5 - Co

mp

ute th

e interval w

idth

How

to Construct a H

istogram

HIS

TOG

RA

MV

IEW

GR

AP

H 11

INTE

RV

AL

NU

MB

ER

STA

RTIN

GV

ALU

EE

ND

ING

VA

LUE

1 2 3 4 5 6 7 8 910

-180

-120

-060

000

060

120

180

240

300

360

INTE

RV

AL

WID

TH

60606060606060606060

-120

-060

000

060

120

180

240

300

360

420

NU

MB

ER

OF

CO

UN

TS

3 5132022242018 6 4

Equal to or greater than the

STA

RTIN

G V

ALU

E

But less than the

EN

DIN

G V

ALU

E

Step

6 - Determ

ine th

e starting

po

int o

f each in

tervalS

tep 7 - C

ou

nt th

e nu

mb

er of p

oin

ts in each

interval

How

to Construct a H

istogram

HIS

TOG

RA

MV

IEW

GR

AP

H 12

US

S C

RO

MM

ELIN

(FFG-37), P

AC

IFIC M

ISS

ILE FIR

ING

RA

NG

E, 135 B

L&P

RO

UN

DS

/MO

UN

T 31, 25 JUN

E 94

LEG

EN

D:

Step

8 - Plo

t the d

ata S

tep 9 - A

dd

the title an

d leg

end

How

to Construct a H

istogram

0 5 10

15

20

25-180

-120-060

000060

120180

240300

360420

TAR

GE

T

YA

RD

S LO

NG

YA

RD

S S

HO

RT

MIS

S D

ISTA

NC

E FO

R M

K 75 G

UN

TES

T FIRIN

GH

ITSM

ISS

ES

SHOTCOUNT

MIS

SE

S

HIS

TOG

RA

MV

IEW

GR

AP

H 13

TargetTarget

TargetTarget

Interpreting Histogram

s L

ocatio

n an

d S

pread

of D

ata

A

DC

B

HIS

TOG

RA

MV

IEW

GR

AP

H 14


sIs P

rocess W

ithin

Sp

ecification

Lim

its?

TargetLS

LU

SL

LSL

US

LTarget

WITH

IN LIM

ITSO

UT

OF

SP

EC

LSL = Low

er specification limit

US

L = Upper specification lim

it

HIS

TOG

RA

MV

IEW

GR

AP

H 15


sP

rocess V

ariation

Target

Day 1

Target

Day 2

Target Day 3

Day 4

Target

HIS

TOG

RA

MV

IEW

GR

AP

H 16

Skew

ed(not sym

metrical)

Discontinued


sC

om

mo

n H

istog

ram S

hap

es

Sym

metrical

(mirror im

aged)

HIS

TOG

RA

MV

IEW

GR

AP

H 17

Step

1 - Co

un

t the n

um

ber o

f data p

oin

ts

WO

RK

SH

EE

T

TOTA

L NU

MB

ER

=

HIS

TOG

RA

MV

IEW

GR

AP

H 18

WO

RK

SH

EE

T

Step

2 - Su

mm

arize the d

ata on

a tally sheet

VA

LUE

TALLY

VA

LUE

TALLY

VA

LUE

TALLY

VA

LUE

TALLY

VA

LUE

TALLY

HIS

TOG

RA

MV

IEW

GR

AP

H 19

Larg

est value

=_______________

Sm

allest value

=_______________

________________________________________

Ran

ge o

f values

=_______________

Step

3 - Co

mp

ute th

e rang

e for th

e data set

WO

RK

SH

EE

T

HIS

TOG

RA

MV

IEW

GR

AP

H 20

Step

4 - Determ

ine th

e nu

mb

er of in

tervals

WO

RK

SH

EE

T

IF Y

OU

HA

VE

TH

ISM

AN

Y D

AT

A P

OIN

TS

US

E T

HIS

NU

MB

ER

OF

INT

ER

VA

LS:

Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals

HIS

TOG

RA

MV

IEW

GR

AP

H 21

Step

5 - Co

mp

ute th

e interval w

idth

WO

RK

SH

EE

T

IntervalW

idth

Range

Num

ber ofIntervals

==Round up to

next higherw

hole number

=

HIS

TOG

RA

MV

IEW

GR

AP

H 22

Step

6 - Determ

ine th

e starting

po

int o

f each in

terval

WO

RK

SH

EE

T

INT

ER

VA

LS

TA

RT

ING

INT

ER

VA

LE

ND

ING

NU

MB

ER

NU

MB

ER

VA

LUE

WID

TH

VA

LUE

OF

CO

UN

TS

12345678910

Step

7 - Co

un

t the n

um

ber o

f po

ints in

each in

terval

HIS

TOG

RA

MV

IEW

GR

AP

H 23

Step

8 - Plo

t the d

ataS

tep 9 - A

dd

title and

legen

d

WO

RK

SH

EE

T

HIS

TOG

RA

MV

IEW

GR

AP

H 24

Step 1 - C

ount the number of data points

EX

ER

CIS

E 1 A

NS

WE

R K

EY

1122

15 7

1320

2512

1619

414

1116

1832

1016

1710

811

2314

1610

521

26 10

2312

1016

1724

1120

913

2410

1618

2215

1319

15 24

1120

1513

918

2216

18 9

1420

1119

1017

1512

1711

1711

1511

1516

1228

1413

TOTA

L = 80

HIS

TOG

RA

MV

IEW

GR

AP

H 25

Step 2 - S

umm

arize the data on a tally sheet

EX

ER

CIS

E 1 A

NS

WE

R K

EY

0 0

10

20

30

41

51

60

71

81

93

107

%

FAT

NO

. OF

PE

RS

11 9

12 4

13 5

14 4

15 7

16 8

17 5

18 4

19 3

20 4

21 1

%

FAT

NO

. OF

PE

RS

22 3

23 2

24 3

25 1

26 1

27 0

28 1

29 0

30 0

31 0

32 1

%

FAT

NO

. OF

PE

RS

HIS

TOG

RA

MV

IEW

GR

AP

H 26

Step 3 - C

ompute the range for the data set

EX

ER

CIS

E 1 A

NS

WE

R K

EY

Larg

est value

=32 P

ercent body fat

Sm

allest value

= 4 P

ercent body fat

_________________________________________

Ran

ge o

f values

=28 P

ercent body fat

HIS

TOG

RA

MV

IEW

GR

AP

H 27

Step 4 - D

etermine the num

ber of intervals

EX

ER

CIS

E 1 A

NS

WE

R K

EY

Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals

IF Y

OU

HA

VE

TH

ISM

AN

Y D

AT

A P

OIN

TS

US

E T

HIS

NU

MB

ER

OF

INT

ER

VA

LS:

HIS

TOG

RA

MV

IEW

GR

AP

H 28

3.5

Step 5 - C

ompute the interval w

idth

EX

ER

CIS

E 1 A

NS

WE

R K

EY

IntervalW

idth

Range

Num

ber ofIntervals

==R

ound upto 4

=288

Use 8 for the num

berof intervals

HIS

TOG

RA

MV

IEW

GR

AP

H 29

EX

ER

CIS

E 1 A

NS

WE

R K

EY

Equal to or greater than

the STA

RTIN

G V

ALU

EB

ut less thanthe E

ND

ING

VA

LUE

Step 6 - D

etermine the starting point of each interval

Step 7 - C

ount the number of points in each interval

INTE

RV

AL

STA

RTIN

GIN

TER

VA

LE

ND

ING

NU

MB

ER

NU

MB

ER

VA

LUE

WID

THV

ALU

EO

F CO

UN

TS1

4+ 4

83

28

+ 412

203

12+ 4

1620

416

+ 420

205

20+ 4

2410

624

+ 428

57

28+ 4

321

832

+ 436

1

HIS

TOG

RA

MV

IEW

GR

AP

H 30

EX

ER

CIS

E 1 A

NS

WE

R K

EY

LEG

EN

D: U

SS

LEA

DE

R (M

SO

-490), 25 JUN

E 94, A

LL 80 PE

RS

ON

NE

L SA

MP

LED

4 8

1216

2024

2832

360

JUN

E 94 P

RT

PE

RC

EN

T B

OD

Y F

AT

SA

TISFA

CTO

RY

% B

OD

Y FA

T

PE

RC

EN

T BO

DY

FAT

NO. OF PERSONNEL

0 4 6 8 10 12 14 16 18 202

Step 8 - P

lot the data S

tep 9 - Add title and legend

HIS

TOG

RA

MV

IEW

GR

AP

H 31

160190

155300

280185

250285

200165

175190

210225

275240

170185

215220

270265

255235

170175

185195

200260

180245

270200

200220

265270

250230

255180

260240

245170

205260

215 185

255245

210225

225235

230230

195225

230255

235195

220210

235240

200220

195235

230215

225235

225200

245230

220215

225250

220245

195235

225230

210240

215230

220225

200235

215240

220230

225215

225

Step 1 - C

ount the number of data points

TOTA

L = 105

EX

ER

CIS

E 2 A

NS

WE

R K

EY

HIS

TOG

RA

MV

IEW

GR

AP

H 32

155 1

160 1

165 1

170 3

175 2

180 2

185 4

190 2

195 5

200 7

205 1

210 4

215 7

220 8

225 11

230 9

235 8

240 5

245 5

250 3

255 4

260 3

265 2

270 3

275 1

280 1

285 1

290 0

295 0

300 1

Step 2 - S

umm

arize the data on a tally sheetE

XE

RC

ISE

2 AN

SW

ER

KE

Y

SC

OR

ETA

LLYS

CO

RE

TALLY

SC

OR

ETA

LLY

HIS

TOG

RA

MV

IEW

GR

AP

H 33

Step 3 - C

ompute the range for the data set

EX

ER

CIS

E 2 A

NS

WE

R K

EY

Larg

est value

=300 P

oints

Sm

allest value

=155 P

oints

__________________________________

Ran

ge o

f values

=145 P

oints

HIS

TOG

RA

MV

IEW

GR

AP

H 34

Step 4 - D

etermine the num

ber of intervals

EX

ER

CIS

E 2 A

NS

WE

R K

EY

Less than 50

50 to 99

100 to 250

More than 250

5 to 7 intervals

6 to 10 intervals

7 to 12 intervals

10 to 20 intervals

IF Y

OU

HA

VE

TH

ISM

AN

Y D

AT

A P

OIN

TS

US

E T

HIS

NU

MB

ER

OF

INT

ER

VA

LS:

HIS

TOG

RA

MV

IEW

GR

AP

H 35

14.5

Step 5 - C

ompute the interval w

idth

EX

ER

CIS

E 2 A

NS

WE

R K

EY

IntervalW

idth

Range

Num

ber ofIntervals

==R

ound upto 15

=145

10

Use 10 for the num

berof intervals

HIS

TOG

RA

MV

IEW

GR

AP

H 36

EX

ER

CIS

E 2 A

NS

WE

R K

EY

Step 6 - D

etermine the starting point of each interval

Step 7 - C

ount the number of points in each interval

Equal to or greater than

the STA

RTIN

G V

ALU

EB

ut less thanthe E

ND

ING

VA

LUE

INTE

RV

AL

STA

RTIN

GIN

TER

VA

LE

ND

ING

NU

MB

ER

NU

MB

ER

VA

LUE

WID

THV

ALU

EO

F CO

UN

TS1

155+ 15

1703

2170

+ 15185

73

185+ 15

20011

4200

+ 15215

125

215+ 15

23026

6230

+ 15245

227

245+ 15

26012

8260

+ 15275

89

275+ 15

2903

10290

+ 15300

1

HIS

TOG

RA

MV

IEW

GR

AP

H 37

155 170 185 200 215 230 245 260 275 290 300

0 5 10 15 20 25 30

SC

OR

ES

NO. OF PERSONNEL

MA

RK

SM

AN

SH

IP S

CO

RE

S F

OR

9mm

PIS

TO

L

LEG

EN

D: M

CB

H K

AN

EO

HE

BA

Y, H

I; AV

ER

AG

E O

F 4 SC

OR

ES

PE

R C

LAS

S, 105 C

LAS

SE

S, 1 JU

NE

94 - 15 JULY

94

EX

ER

CIS

E 2 A

NS

WE

R K

EY

Step 9 - A

dd title and legend

Step 8 - P

lot the data

Documents

Module 11 - Histogram