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Basic Tools for Process Improvement

Module 11

HISTOGRAM

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What is a Histogram?

A Histogram is a vertical bar chart that depicts the distribution of a set of da

Run Charts or Control Charts, which are discussed in other modules, a Hisdoes not reflect process performance over time. It's helpful to think of a Hisas being like a snapshot, while a Run Chart or Control Chart is more like a

(Viewgraph 1).

When should we use a Histogram?When you are unsure what to do with a large set of measurements presenttable, you can use a Histogram to organize and display the data in a more

friendly format. A Histogram will make it easy to see where the majority of falls in a measurement scale, and how much variation there is. It is helpful construct a Histogram when you want to do the following (Viewgraph 2):

! Summarize large data sets graph ically. When you look at Viewgyou can see that a set of data presented in a table isnt easy to use. make it much easier to understand by summarizing it on a tally shee

(Viewgraph 7) and organizing it into a Histogram (Viewgraph 12).

! Compare process results with specification limits. If you add

process specification limits to your Histogram, you can determine quwhether the current process was able to produce "good" products.

Specification limits may take the form of length, weight, density, quanmaterials to be delivered, or whatever is important for the product of

process. Viewgraph 14 shows a Histogram on which the specificatioor "goalposts," have been superimposed. Well look more closely atimplications of specification limits when we discuss Histogram interp

later in this module.

! Communicate information g raphically. The team members cansee the values which occur most frequently. When you use a Histogsummarize large data sets, or to compare measurements to specific

limits, you are employing a powerful tool for communicating informat

! Use a tool to assist in decision making. As you will see as we m

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HISTOGRAM VIEWG

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 specification

Communicate information to the team


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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 t

Histogram.

2. Horizontal or X-Axis : The horizontal or X-axis shows you the sca

values into which the measurements fit. These measurements are ggrouped into intervals to help you summarize large data sets. Individpoints are not displayed.

3. Bars: The bars have two important characteristicsheight and widt

height represents the number of times the values within an interval oThe width represents the length of the interval covered by the bar. It

same for all bars.

4. Vertical or Y-Axis : The vertical or Y-axis is the scale that shows y

number of times the values within an interval occurred. The number

is also referred to as "frequency."

5. Legend: The legend provides additional information that documents

the data came from and how the measurements were gathered.

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HISTOGRAM VIEWG

1 Title 2 Horizon tal / X-axis

3 Bars 4 Vertical / Y-axis

5 Legend

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

DAYS OF OPERATION PRIOR TO

FAILURE FOR AN HF RECEIVER

DAYS OF OPERATION

MEAN TIME BETWEEN FAILURE (IN DAYS) FOR R-1051 HF RECEIVER

Data taken at SIMA, Pearl Harbor, 15 May - 15 July 94

Parts of a Histogram

1

F

R

E

Q

UE

N

C

Y

4

5

0

20

40

60

80

100


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How is a Histogram constructed?

There are many different ways to organize data and build Histograms. You

safely use any of them as long as you follow the basic rules. In this moduleuse the nine-step approach (Viewgraphs 4 and 5) described on the followin

EXAMPLE: The following scenario will be used as an example to provide

we 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 airbornespotter provided accurate rake data to assess the fall of shotboth 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

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HISTOGRAM VIEWG

Step 1 - Count number of data point

Step 2 - Summarize on a tally sheet

Step 3 - Compute the range

Step 4 - Determine number of interv

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


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Step 1 - Count the total number of data points you have listed. Sup

team collected data on the miss distance for the gunnery exercise descthe example. The data you collected was for the fall of shot both long athe target. The data are displayed in Viewgraph 6. Simply counting the

number of entries in the data set completes this step. In this example, t135 data points.

Step 2 - Summarize your data on a tally sheet. You need to summariz

data to make it easy to interpret. You can do this by constructing a tally

First, identify all the different values found in Viewgraph 6 (-160, -010

220, etc.). Organize these values from smallest to largest (-180, -12410).

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

Alternatively, simply count the number of times each value is presendata set and enter that number next to the value, as shown in Viewg

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

intervals of values.

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HISTOGRAM VIEWG

Step 1 - Count the total number of data po

-180 30 190 380 330 140 160 270 10 - 9- 10 30 60 230 90 120 10 50 250 18

-130 220 170 130 - 50 - 80 180 100 110 20260 190 -100 150 210 140 -130 130 150 37160 180 240 260 - 20 - 80 30 80 240 13210 40 70 - 70 250 360 120 - 60 - 30 20

50 20 30 280 410 70 - 10 20 130 17140 220 - 40 290 90 100 - 30 340 20 8210 130 350 250 - 20 230 180 130 - 30 21-30 80 270 320 30 240 120 100 20 7

300 260 20 40 - 20 250 310 40 200 19

110 -30 50 240 180 50 130 200 280 6260 70 100 140 80 190 100 270 140 8110 130 120 30 70

TOTAL = 135

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

How to Construct a Histogram

Step 2 - Summarize the data on a tally she


DATA TALLY DATA TALLY DATA TALLY DATA TALLY DATA TA

- 180 3 90

- 130 2 100- 100 2 110

- 90 5 120

- 80 6 130

- 70 3 140

- 60 4 150

1

21

1

2

1

1

- 20

- 1010

20

30

40

50

2

53

4

8

5

2

190

200210

220

230

240

250

4

44

2

2

4

4

290

300310

320

330

340

350


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Step 3 -Compute the range for the data set. Compute the range by su

the smallest value in the data set from the largest value. The range repthe extent of the measurement scale covered by the data; it is always a number. The range for the data in Viewgraph 8 is 590 yards. This num

obtained by subtracting -180 from +410. The mathematical operation bdown in Viewgraph 8 is:

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

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

Step 4 - Determine the number of intervals required . The number of

influences the pattern, shape, or spread of your Histogram. Use the foll

table (Viewgraph 9) to determine how many intervals (or bars on the bayou should use.

If you have this Use this number many 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 inte

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HISTOGRAM VIEWG

Largest value = + 410 yards past target

Smallest value = - 180 yards short of targ

Range of values = 590 yards

Step 3 - Compute the range for the data s


Calculation: + 410 - (- 180) = 410 + 180 = 5

IF YOU HAVE THIS

MANY DATA POINTS

USE THIS NUMBER

OF INTERVALS:

Less than 50

50 to 99

5 to 7 intervals

6 to 10 intervals

Step 4 - Determine the number of interva

required



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Step 5 - Compute the interval wid th . To compute the interval width (Vie

10), divide the range (590) by the number of intervals (10). When com

interval width, you should round the data up to the next higher whole nu

come up with values that are convenient to use. For example, if the ranis 17, and you have decided to use 9 intervals, then your interval width i

You can round this up to 2.

In this example, you divide 590 yards by 10 intervals, which gives an int

width of 59. This means that the length of every interval is going to be 5

To facilitate later calculations, it is best to round off the value representinwidth of the intervals. In this case, we will use 60, rather than 59, as the

width.

Step 6 - Determine the starting point for each interval. Use the smal

point in your measurements as the starting point of the first interval. Th

point for the second interval is the sum of the smallest data point and thwidth. For example, if the smallest data point is -180, and the interval w

the starting point for the second interval is -120. Follow this procedure(Viewgraph 11) to determine all of the starting points (-180 + 60 = -120; = -60; etc.).

Step 7 - Count the number of poin ts that fall within each in terval. T

the data points that are equal to or greater than the starting value and

the ending value (also illustrated in Viewgraph 11). For example, if the

interval begins with -180 and ends with -120, all data points that are equgreater than -180, but still less than -120, will be counted in the first intein mind that EACH DATA POINT can appear in only one interval.

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HISTOGRAM VIEWG

59Interval

Width

Range

Number ofIntervals

=590

10

=

Use 10 for the

number of intervals

Use 10 for the

number of intervals

Round u

to 60

=

Step 5 - Compute the interval width


INTERVAL

NUMBER

STARTING

VALUE

ENDING

VALUE

1

2

3

4

5

6

-180

-120

-060

000

060

120

INTERVAL

WIDTH

60

60

60

60

60

60

-120

-060

000

060

120

180

2 0

NUMBER OF

COUNTS

3

5

13

20

22

24

Step 6 - Determine the starting point of each inte

Step 7 - Count the number of points in each inte



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Step 8 - Plot the data. A more precise and refined picture comes into vie

you plot your data (Viewgraph 12). You bring all of the previous steps towhen you construct the graph.

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

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

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

! The height is determined by the number of observations or percenta

total observations for each of the intervals.

! The Histogram may not be perfectly symmetrical. Variations will occ

yourself whether the picture is reasonable and logical, but be careful

your preconceived ideas influence your decisions unfairly.

Step 9 - Add the title and legend. A title and a legend provide the who,

when, where, and why (also illustrated in Viewgraph 12) that are importaunderstanding and interpreting the data. This additional information docthe nature of the data, where it came from, and when it was collected. T

may include such things as the sample size, the dates and times involve

collected the data, and identifiable equipment or work groups. It is impoinclude any information that helps clarify what the data describes.

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HISTOGRAM VIEWG

USS CROMMELIN (FFG-37), PACIFIC MISSILE FIRING RANGE, 135 BL&P ROUNDS/MOUNT 31, 2LEGEND:

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 FIRING

HITS

MISSESSHO

T

COUNT

MISSES


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How do we interpret a Histogram?

A Histogram provides a visual representation so you can see where most o

measurements are located and how spread out they are. Your Histogram mshow 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 awthe target, as in Viewgraph 13B.

! Even when most of the data were close together, they were located target 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 cowas. You may be thinking, "So what? How will this help me do my job bett

with the results of the process clearly depicted, we can find the answer to aquestion:

Did the process produce goods and services which are within specificatio

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

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HISTOGRAM VIEWG

Target Target

Target Tar

Interpreting HistogramsLocation and Spread of Data

A

DC

B

Interpreting HistogramsIs Process Within Specif ication

Limits?

WITHIN LIMITS OUT OF SPEC


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Portraying your data in a Histogram enables you to check rapidly on the nuthe percentage, of defects produced during the time you collected data. Bu

you know whether the process was stable (Viewgraph 15), you wont be apredict whether future products will be within specification limits or determin

course of action to ensure that they are.

A Histogram can show you whether or not your process is producing produ

services that are within specification limits. To discover whether the processtable, and to predict whether it can continue to produce within spec limits,

to 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 aappropriate course of actionto eliminate special causes of variation, or tofundamental changes to your process.

There are times when a Histogram may look unusual to you. It might have one peak, be discontinued, or be skewed, with one tail longer than the othe

shown in Viewgraph 16. In these circumstances, the people involved in the

should ask themselves whether it really is unusual. The Histogram may nosymmetrical, but you may find out that it should look the way it does. On t

hand, the shape may show you that something is wrong, that data from sev

sources were mixed, for example, or different measurement devices were uoperational definitions weren't applied. What is really important here is to a

jumping to conclusions without properly examining the alternatives.

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HISTOGRAM VIEWG

Interpreting HistogramsProcess Variation

Target

Day 1

Target

Day 2

Target

Day 3 Day 4

Target

Skewed

(not symmetrical)

Discontinued

Interpreting HistogramsCommon Histogram Shapes


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How can we practice what we've learned?

Two exercises are provided that will take you through the nine steps for dev

Histogram. On the four pages that follow the scenario for Exercise 1 you wset of blank worksheets (Viewgraphs 17 through 23) to use in working throuof the exercises in this module.

You will find a set of answer keys for Exercise 1 after the blank worksheetsExercise 2 after the description of its scenario. These answer keys represe

one possible set of answers. It's all right for you to choose an interval widthnumber of intervals that is different from those used in the answer keys. Evthough the shape of your Histogram may vary somewhat from the answer k

shape, it should be reasonably close unless you used a very different numbintervals.

EXERCISE 1: The source of data for the first exercise is the following sce

list of the data collected follows this description. Use the blank worksheetsViewgraphs 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 forpercent body fat. Prior

to one PRT, the corpsman recorded the percent of body fatfor the 80 personnel assigned to the command. These are

the data collected:

PERCENT BODY FAT RECORDED

11 22 15 7 13 20 25 12 16

4 14 11 16 18 32 10 16 17

8 11 23 14 16 10 5 21 26

23 12 10 16 17 24 11 20 9

24 10 16 18 22 15 13 19 15

11 20 15 13 9 18 22 16 18

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HISTOGRAM VIEWG

Step 1 - Count the number of data pointWORKSHEET

TOTAL NUMBER =

WORKSHEET


VALUE TALLY VALUE TALLY VALUE TALLY VALUE TALLY VALUE T


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HISTOGRAM VIEWGR

Largest value = _______________

Smallest value = _______________

________________________________________

Range of values = _______________

Step 3 - Compute the range for the data se

WORKSHEET

Step 4 - Determine the number of intervals

WORKSHEET

IF YOU HAVE THIS

MANY DATA POINTS

USE THIS NUMBER

OF INTERVALS:

Less than 50

50 to 99

5 to 7 intervals

6 to 10 intervals


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HISTOGRAM VIEWG


WORKSHEET

Interval

Width

Range

Number ofIntervals

= =

Round up

next high

whole num

=

Step 6 - Determine the starting point of each inteWORKSHEET

INTERVAL STARTING INTERVAL ENDING NUMBER

NUMBER VALUE WIDTH VALUE OF COUNTS

1

2

3

4

5

Step 7 - Count the number of points in each in ter


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HISTOGRAM VIEWGR


Step 9 - Add ti tle and legend

WORKSHEET


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HISTOGRAM VIEWG

Step 1 - Count the number of data points

EXERCISE 1 ANSWER KEY

11 22 15 7 13 20 25 12 16 1

4 14 11 16 18 32 10 16 17 1

8 11 23 14 16 10 5 21 26 1

23 12 10 16 17 24 11 20 9 1

24 10 16 18 22 15 13 19 15 2

11 20 15 13 9 18 22 16 18

14 20 11 19 10 17 15 12 17 1

17 11 15 11 15 16 12 28 14 1

TOTAL = 80



0 0

1 02 0

3 0

4 1

5 1

6 0

%FAT NO. OF PERS

11 9

12 413 5

14 4

15 7

16 8

17 5

%FAT NO. OF PERS

22 3

23 224 3

25 1

26 1

27 0

28 1

%FAT NO. OF P


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HISTOGRAM VIEWGR

Step 3 - Compute the range for the data set


Largest value = 32 Percent body fat

Smallest value = 4 Percent body fat

_________________________________________

Range of values = 28 Percent body fat

Step 4 - Determine the number of intervals


Less than 50

50 to 99

5 to 7 intervals

6 to 10 intervals

IF YOU HAVE THIS

MANY DATA POINTS

USE THIS NUMBER

OF INTERVALS:


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EXERCISE 1 ANSWER KEYStep 6 - Determine the starting point of each inter

Step 7 - Count the number of points in each interv



1 4 + 4 8 3

2 8 + 4 12 20

3 12 + 4 16 20

4 16 + 4 20 20

5 20 + 4 24 10

HISTOGRAM VIEWG

3.5



Interval

Width

Range

Number ofIntervals

= =

Round u

to 4

=

28

8

Use 8 for the number

of intervals


B i T l f P I t

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HISTOGRAM VIEWGR


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.OFPERSONNEL

0

4

6

8

10

12

14

16

18

20

2


Step 9 - Add ti tle and legend



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EXERCISE 2: The source of data for the second exercise is the followin

A listing of the data collected follows this description. Use the blank work

Viewgraphs 17 through 23 to do this exercise. You will find answer keys Viewgraphs 31 through 37.

A Marine Corps small arms instructor was performing ananalysis of 9 mm pistol marksmanship scores to improve

training methods. For every class of 25, the instructorrecorded the scores for each student who occupied thefirst four firing positions at the small arms range. The

instructor 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

175 190 210 225 275 240 170 185 215

270 265 255 235 170 175 185 195 200

180 245 270 200 200 220 265 270 250

255 180 260 240 245 170 205 260 215

255 245 210 225 225 235 230 230 195

230 255 235 195 220 210 235 240 200

195 235 230 215 225 235 225 200 245

220 215 225 250 220 245 195 235 225

210 240 215 230 220 225 200 235 215

220 230 225 215 225


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HISTOGRAM VIEWGR

160 190 155 300 280 185 250 285 200 16

175 190 210 225 275 240 170 185 215 22

270 265 255 235 170 175 185 195 200 26

180 245 270 200 200 220 265 270 250 23255 180 260 240 245 170 205 260 215 18

255 245 210 225 225 235 230 230 195 22

230 255 235 195 220 210 235 240 200 22

195 235 230 215 225 235 225 200 245 23

220 215 225 250 220 245 195 235 225 23

210 240 215 230 220 225 200 235 215 24

220 230 225 215 225

Step 1 - Count the number of data points

TOTAL = 105


155 1

160 1

165 1

170 3

175 2

180 2

205 1

210 4

215 7

220 8

225 11

230 9

255 4

260 3

265 2

270 3

275 1

280 1

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

SCORE TALLY SCORE TALLY SCORE TALLY



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HISTOGRAM VIEWG

Step 3 - Compute the range for the data se


Largest value = 300 Points

Smallest value = 155 Points

__________________________________

Range of values = 145 Points

Step 4 - Determine the number of interval


Less than 50

50 to 99

5 to 7 intervals

6 to 10 intervals

IF YOU HAVE THIS

MANY DATA POINTS

USE THIS NUMBER

OF INTERVALS:



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Step 6 - Determine the starting point of each interv

Step 7 - Count the number of points in each interv



1 155 + 15 170 3

2 170 + 15 185 7

3 185 + 15 200 11

4 200 + 15 215 12

5 215 + 15 230 26

HISTOGRAM VIEWGR

14.5



Interval

Width

Range

Number ofIntervals

= =

Round up

to 15

=

145

10

Use 10 for the number

of intervals



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HISTOGRAM VIEWG

155 170 185 200 215 230 245 260 275 290 300

0

5

10

15

20

25

30

SCORES

NO.OFPERSONN

EL

MARKSMANSHIP SCORES FOR 9mm PISTOL

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


Step 9 - Add title and legend


p


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p

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 QuaLeadership (Instructor Guide), pp. 6-44 - 6-47. San Diego, CA: Navy PResearch and Development Center.

3. Department of the Navy (September 1993). Systems Approach to ProcImprovement (Instructor Guide), pp. 10-17 - 10-38. San Diego, CA: OU

Quality Leadership Office and Navy Personnel Research and DevelopmCenter.

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

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HISTOGRAM

VIEWGRAPH1

W

hatIsa

Histogram?

Abargra

phthatshowsthedistributionofdata

Asnapsh

otofdatatak

enfromapro

cess

0

5

10

15

20

25

30

35

40

45

50

55

60

020

40

60

80

100

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HISTOGRAM

VIEWGRAPH2

When

AreHistograms

Used?

Summarizelargedatasetsgrap

hically

Compare

measurem

entstospe

cifications

Communicateinform

ationtothe

team

Assistindecisionmaking

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HISTOGRAM

VIEWGRAPH3

1

Title

2Horizontal/X-axis

3

Bars

4

Vertical/Y-axis

5

Legend

0

5

10

1

5

20

25

30

35

40

45

50

55

60

DAY

SOF

OPERATIONPRIORT

O

FAILUREF

ORANH

FRE

CEIVER

DAY

SOF

OPE

RATION

MEANTIMEBE

TWEENFAIL

URE

(IND

AYS)F

ORR-1

051HFRECEIVER

Datataken

atSIMA

,P

earlH

ar

bor,1

5M

ay-1

5July

94

PartsofaHistogra

m

1

32

FREQUENCY

45

0 204

060

80

100

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HISTOGRAM

VIEWGRAPH4

Step1-

Coun

tn

um

ber

of

da

tapoin

ts

Step2-

Summ

ariz

eon

atall

ysh

eet

Step3-

Computetheran

ge

Step4-D

etermin

e

num

ber

ofin

terv

al

s

Step5-

Computeinterv

alwi

dth

Cons

tructing

aHisto

gram

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HISTOGRAM

VIEWGRAPH5

Cons

tructing

aHisto

gram

Step6-

Determin

einterv

al

startin

g

points

Step7-

Coun

tn

u

mber

of

pointsin

eachin

te

rval

Step8-

Plotthed

ata

Step9-

Addtitleandle

gen

d

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HISTOGRAM

VIEWGRAPH6

Step1-Cou

ntthetotalnumberofdatapoints

-18

0

30

190

380

330

1

40

160

27

0

1

0

-90

-1

0

30

60

230

90

1

20

1

0

50

250

180

-13

0

22

0

17

0

130

-50

-

80

180

100

11

0

200

26

0

190

-100

150

21

0

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