SPC tools and techniques– part 2 & Fundamentals of Statistics—Part 1 1. In the article The tools of Quality Part IV: Histograms (Quality Prog

SPC tools and techniques– part 2 & Fundamentals of Statistics—Part 1
1. In the article The tools of Quality Part IV: Histograms (Quality Progress, September 1990, Vol. XXIII, No.9, pp.75-78) data are presented on the gain of 120 amplifiers. These data are reproduced below.
a) Construct a stem-and-leaf display step by step. Category Interval=1 (7,8,9,10,11 as stems, and decimal number as leaf) (3pts).
b) Comment on the shape of the display in a. Can you make an assumption that the data follows normal distribution? Why? (2pt)
c) Construct a stem-and-leaf display step by step. Category Interval=0.5 (7.0X<7.5, 7.5X<8, 8.0X<8.5, .) (1pts). 8.1 10.4 8.8 9.7 7.8 9.9 11.7 8.0 9.3 9.0 8.2 8.9 10.1 9.4 9.2 7.9 9.5 10.9 7.8 8.3 9.1 8.4 9.6 11.1 7.9 8.5 8.7 7.8 10.5 8.5 11.5 8.0 7.9 8.3 8.7 10.0 9.4 9.0 9.2 10.7 9.3 9.7 8.7 8.2 8.9 8.6 9.5 9.4 8.8 8.3 8.4 9.1 10.1 7.8 8.1 8.8 8.0 9.2 8.4 7.8 7.9 8.5 9.2 8.7 10.2 7.9 9.8 8.3 9.0 9.6 9.9 10.6 8.6 9.4 8.8 8.2 10.5 9.7 9.1 8.0 8.7 9.8 8.5 8.9 9.1 8.4 8.1 9.5 8.7 9.3 8.1 10.1 9.6 8.3 8.0 9.8 9.0 8.9 8.1 9.7 8.5 8.2 9.0 10.2 9.5 8.3 8.9 9.1 10.3 8.4 8.6 9.2 8.5 9.6 9.0 10.7 8.6 10.0 8.8 8.6 Please be noted: you are not expected to use Excel Histogram function in this problem. In the next assignment, this function must be used. 2. An experiment in chemistry looked at the effect of temperature on the solubility of salt in water. Below are data on the solubility of Potassium Chloride (KCL). Construct a scatter plot of the data and comment on the relationship between temperature and solubility using Microsoft Excel. (3pts) Temp. C 0 10 20 30 40 50 60 70 80 90 100 Solubility 29.6 28.0 33.6 38.1 34.2 42.6 44.8 48.1 56.5 55.4 62.9 1. In the article The tools of Quality Part IV: Histograms (Quality Progress, September 1990, Vol. XXIII, No.9, pp.75-78) data are presented on the gain of 120 amplifiers. These data are reproduced below. a) Construct a stem-and-leaf display step by step. Category Interval=1 (7,8,9,10,11 as stems, and decimal number as leaf) (3pts). b) Comment on the shape of the display in a. Can you make an assumption that the data follows normal distribution? Why? (2pt) c) Construct a stem-and-leaf display step by step. Category Interval=0.5 (7.0X<7.5, 7.5X<8, 8.0X<8.5, .) (1pts). 8.1 10.4 8.8 9.7 7.8 9.9 11.7 8.0 9.3 9.0 8.2 8.9 10.1 9.4 9.2 7.9 9.5 10.9 7.8 8.3 9.1 8.4 9.6 11.1 7.9 8.5 8.7 7.8 10.5 8.5 11.5 8.0 7.9 8.3 8.7 10.0 9.4 9.0 9.2 10.7 9.3 9.7 8.7 8.2 8.9 8.6 9.5 9.4 8.8 8.3 8.4 9.1 10.1 7.8 8.1 8.8 8.0 9.2 8.4 7.8 7.9 8.5 9.2 8.7 10.2 7.9 9.8 8.3 9.0 9.6 9.9 10.6 8.6 9.4 8.8 8.2 10.5 9.7 9.1 8.0 8.7 9.8 8.5 8.9 9.1 8.4 8.1 9.5 8.7 9.3 8.1 10.1 9.6 8.3 8.0 9.8 9.0 8.9 8.1 9.7 8.5 8.2 9.0 10.2 9.5 8.3 8.9 9.1 10.3 8.4 8.6 9.2 8.5 9.6 9.0 10.7 8.6 10.0 8.8 8.6 Please be noted: you are not expected to use Excel Histogram function in this problem. In the next assignment, this function must be used. 2. An experiment in chemistry looked at the effect of temperature on the solubility of salt in water. Below are data on the solubility of Potassium Chloride (KCL). Construct a scatter plot of the data and comment on the relationship between temperature and solubility using Microsoft Excel. (3pts) Temp. C 0 10 20 30 40 50 60 70 80 90 100 Solubility 29.6 28.0 33.6 38.1 34.2 42.6 44.8 48.1 56.5 55.4 62.9 1 1 Lecture 7: SPC tools and techniques part 2 Chapter 4 2 Magnificent seven quality tools Process flow diagram Pareto diagram Cause and effect diagram Check sheet Scatter diagram Histogram Control charts (include run chart) 3 Check sheet for paint non-conformities 4 Check sheet for plastic mold nonconformities 5 Check sheet example pin diameter 6 Check sheet example-- for car door painting process The Quality Toolbook from http://syque.com/quality_tools/toolbook/Check/example.htm 2 7 Key points in using check sheet Check sheet should be well designed, easy to read, and clearly labeled. Record only necessary information. Dont attempt to collect data not specifically related to the issues being studied. Keep it simple 8 Scatter diagram (scatterplot) Display the relationship between two variables Gas mileage and speed Cutting speed and tool life Feed rate and surface roughness 9 Constructing scatter diagram Two measurements on each unit Plot using coordinate axes Vertical: response Horizontal: explanatory or predictor Explanatory variable R e s p o n s e v a ri a b le 10 Interpretation-- scatter plot Positive Negative No association Curvilinear 11 Interpretation-- scatter plot 12 Scatter diagram example 3 13 Scatter diagram example Fig. 22-17, cutting speed and tool life, from Materials and Processes in Mfg, by DeGarmo, Black, & Kohser, 9th ed 14 Scatter diagram example 15 Scatter diagram examples 16 Histogram Pictorial representation (summary) of a set of data measurement fr e q u e n c y 17 Interpret histograms By graphic (qualitative) features Symmetry Mounted, flat Skew Right, left By analytical (quantitative) features Center: mean, medium, mode Dispersion: range, standard deviation 18 symmetric Skew- right Skew- left 4 19 Histogram 20 Constructing a stem-leaf chart (one type of histogram) Order data from smallest to largest Determine cell intervals equal width Construct a frequency table Draw bar chart Vertical: frequency by % Horizontal: cell intervals 21 How many intervals (cells, categories) should be formed? Number of cells is based on judgment General rule of thumb is the number of cells should be between 5 and 20 5 to 9 when observations are < 100 8 to 17 observations are between 100 & 500 15 to 20 when observations are > 500

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Stem-and-leaf chart– octane ratings

The following data on motor octane
ratings are excerpted from an article in
Technometrics Vol.19 p425

93.3 91.8 92.3 90.4 90.1 93.0 88.7 89.9

89.8 89.6 87.4 88.4 88.9 91.2 89.3 94.4

92.7 91.8 91.6 90.4 91.1 92.6 89.3 90.6

91.1 90.4 89.3 89.7 90.3 91.6 90.5 93.7

92.7 92.2 92.2 91.2 91.0 92.2 90.0 90.7

23

Stem-and-leaf chart– octane ratings
86

87 4

88 4 9 7

89 8 6 3 7 3 3 9

90 4 4 4 1 3 5 0 6 7

91 1 8 8 6 2 1 0 2 6

92 7 7 2 3 2 6 2

93 3 0 7

94 4

95 24

Stem-and-leaf chart– octane ratings

86

87 4

88 4 7 9

89 3 3 3 6 7 8 9

90 0 1 3 4 4 4 5 6 7

91 0 1 1 2 2 3 6 8 8

92 2 2 2 3 6 7 7

93 0 3 7

94 4

95

5

25

Octane Rating frequency table

Class interval Freq. Freq.% Cumu. %

87.0Y<88.0 1 2.5 2.5 88.0Y<89.0 3 7.5 10 89.0Y<90.0 7 17.5 27.5 90.0Y<91.0 9 22.5 50 91.0Y<92.0 9 22.5 72.5 92.0Y<93.0 7 17.5 90 93.0Y<94.0 3 7.5 97.5 94.0Y<95.0 1 2.5 100 40 100 26 Histogramoctane rating Histogramoctane rating (created from Excel) 27 1 3 7 9 9 7 3 1 0 1 2 3 4 5 6 7 8 9 10 Frequency Frequency Linear (Frequency) 28 Cautions (limitations) Choice of intervals can affect the picture you get Histograms can hide trends over time or other patterns 29 Run chart A graphic representation of process performance data tracked over time 30 6 31 Run chart 32 Control chart A graphic display of the results of a process over time and against established control limits Control chart Run chart with control limits A statistical tool used to detect excessive process variability due to specific assignable causes that can be corrected. Two purposes To see if special causes are present To test or evaluate if a solution works 33 Control chart Assignment today InClass Practice 5 34 1 Lecture 8: Fundamentals of StatisticsPart 1 Chapter 5 Outline Statistics, type of data, data description Frequency distribution and histogram Measures of central tendency Measures of dispersion Population, sample, Normal curve Computer program Types of Data Variable Quality characteristics that are measurable, continuous. Example: length, weight Attribute Classified either conforming or nonconforming (yes/no, accept/reject) Go/NoGo gauge A Go/no go refers to an inspection tool used to check a workpiece against its allowed tolerances. Its name derives from its use: the gauge itself has two tests; the check involves the workpiece's having to pass one test (Go) and 'fail' the other (No Go). Go/NoGo gauge Lower gauge is a plain plug gauge of 12.60mm for 'Go' and 12.90mm as 'NoGo For checking tolerance between 12.60-12.90. For a washer with dia= 12.75, pass one and fail the other and a washer with dia= 12.5, fail at both end Describe Data Summarizing data: Graphical (describe picture) Plot or picture of frequency distribution Analytical (math calculations) Measures of central tendency (mean, mode, median) Measures of dispersion (range, std. deviation) http://www.answers.com/topic/go-no-go http://www.answers.com/topic/engineering-tolerance http://upload.wikimedia.org/wikipedia/commons/2/26/GaugePlugThreadGoNoGo.jpg 2 Descriptive statistics Mean = average() Median = median() Mode = Mode() Std. Dev. = STDEV() Variance = var() Count = count() Max = max() Min = min() Range = max()-min() Skewness = skew() Kurtosis = kurt() Sum =sum() Student scores 82.64, 84.47, 77.12, 79.08, 95.21, 81.84, 94.66, 80.50, 84.96, 89.18, 87.76 Mean=85.22 Median=84.47 Max= 95.21 Min=77.12 Describe Data 0 1 2 3 4 5 6 7 <75 <80 <85 <90 <95 <100 Frequency Frequency Frequency Distribution Ungrouped data: a listing of the individual observed values Grouped data: a lumping together (subgroups) of the observed values Frequency Distribution ungrouped data A table that includes the number of daily billing errors of a larger organization Example number of daily billing errors Organize the data (array) Tabulate the frequency of each value Different types of frequency distributions Histogram: consists of a set of rectangles that represent the frequency in each 3 Different types of frequency distributions Relative frequency distribution: proportion or fraction of the total Different types of frequency distributions Cumulative frequency Different types of frequency distributions Relative cumulative frequency Summarization of this billing example This example shows how to organize and sort the data by frequency, relative frequency, cumulative frequency, and relative cumulative frequency. We have practiced this tasks in Pareto chart Students should be able to chart these frequency data by Excel Frequency Distribution grouped data Sometimes the collected data is a large volume of data set. If we still use the previous method, the frequency distribution result may not give a good explanation. Frequency Distribution example of grouped data Example: steel shaft weight (kilograms) Simplify data by coded value: the weights are coded from 2.500Kg, a weight with a value of 31 is equivalent to 2.531Kg (2.500+0.031) 4 Steel shaft weight Total: 110 Highest: 75 Lowest: 31 Frequency Distribution grouped data Collect data and construct a tally sheet Determine the range Determine the cell (subgroup) interval Determine the cell midpoints Determine the cell boundaries Post the cell frequency 1. Data tabulation: from lowest to highest, a tally sheet A large number of categories 2. Determine the range Where R=range Xh= highest data Xl = lowest data 044.0531.2575.2 lh XXR How many cells (categories) should be formed grouping Number of cells is based on judgment General rule of thumb is the number of cells should be between 5 and 20 5 to 9 when observations are < 100 8 to 17 observations are between 100 & 500 15 to 20 when observations are > 500

Three issues in creating the
cells

Equal width of cell intervals

Cell midpoints

Cell boundaries

5

3. Determine the cell interval
width

Cell interval is the distance between adjacent
cell midpoints. Odd number is preferred
because of easy calculation

0057.0
)110log(322.31

044.0

log322.31

n

R
i

3. Determine the cell interval
length

All trial-and-error method, here h is the total
number of intervals needed

Assume i=0.003, then

Assume i=0.005, then

Assume i=0.007, then

9
005.0

044.0

i

R
h

15
003.0

044.0

i

R
h

6
007.0

044.0

i

R
h

4.Determine the cell
boundaries and midpoints

When constructing a histogram, its
important to remember two things:

Histogram must contain all of the data

One particular value cannot fit into two
different cells, which means cells cannot
overlap

4.Determine the cell
boundaries and midpoints

1st method– the simplest technique is
to choose the lowest value measured as
first midpoint, and so on

2nd method– lowest value will be the
lower boundary for first cell (then, the
first midpoint would be the lowest value
plus half of interval), and so on

2

i
XMP

ll

4.Determine the cell boundaries and
midpoints by 1st method

Lowest value as the first
midpoint so

first Midpoint =2.531

Since width=0.005, so first
cell boundaries:

Lower-B=2.531-0.005/2

= 2.5285

Upper-B=2.531+0.005/2

=2.5335

Midpoint Cell boundaries

2.531 2.5285 – 2.5335

2.536 2.5335 – 2.5385

2.541 2.5385 2.5435

2.546 2.5435 2.5485

2.551 2.5485 – 2.5535

2.556 2.5535 – 2.5585

2.561 2.5585 2.5635

2.566 2.5635 2.5685

2.571 2.5685 2.5735

2.576 2.5735 2.5785

4.Determine the cell boundaries and
midpoints by 2st method

Lowest value as the lower
boundary of the first cell so
Lower-B=2.531,

Adjust a little bit, then

Lower-B=2.530

First Midpoint =2.530+0.005/2
= 2.5325

Upper-B=2.530+0.005

=2.535

Midpoint Cell boundaries

2.5325 2.530 – 2.535

2.5375 2.535 – 2.540

2.5425 2.540 – 2.545

2.5475 2.545 2.550

2.5525 2.550 – 2.555

2.5575 2.555 – 2.560

2.5625 2.560 2.565

2.5675 2.565 2.570

2.5725 2.570 2.575

6

4.Determine the cell
boundaries and midpoints

Other midpoints: 2.533+0.005=2.538

2.538+0.005=2.543..

533.2
2

005.0
531.2

2

i
XMP

ll

4.Determine the cell
boundaries and midpoints

Boundaries are established so there is no question
as to the location of an observation.

2.530 2.535 2.540 2.575 2.545 2.550 2.555 2.560 2.565 2.570

6. Post the cell frequency Excel Histogram function

Histogram command

Input range entire data

Bin range using upper boundary to
separate the cells

Given these two parameters, Excel will
generate frequency column

Be careful: you must have Data
Analysis ToolPak loaded

Characteristics of Graphs

Symmetry or lack of symmetry of the
data (Distribution around the central
value, Skewness)

Number of peaks

Peakedness of the data

Platykurtic (flat shape)

Leptokurtic (clear peak)

Characteristics of frequency
distributions

7

Analysis of Histogram

Many cases are expected to have a normal
pattern (bell)

Graphical representation can help figure
out problems if there is any in the process.

If not normally distributed, then something
in the process may be out of control or the
data follows some other basic pattern

Differences due to location,
spread, and shape

39

Histogram Computing function
with Excel

Define the data range

Find out mean, median, mode, standard
deviation

Construct histogram

Name data or other names
Use functions to describe data
Define midpoint and U-boundary
Tool Data AnalysisHistogram

Input range data
Bin range U-boundary
Output range

Chart data midpoint and frequency columns
One useful website about histogram with

Excel

Procedures of using Excel to
make histogram

How many cells

or categories

Boundaries of

cells or categories

Excel histogram

function (data, Bin)
Chart

function

R=max(data)-min(data)

i=R/(1+3.322log(n))
# of cells = R/i

The very left cell,

L-bound

U-bound= L-bound + i

U-bound (n+1) = U-

bound (n) +i

Cell

midpoint
Midpoint=(L-bound + U-bound)/2

Midpoint=L-bound + i/2

Assignment today/this week

InClass 6: Histogram function in Excel

Homework 2

http://www.utexas.edu/its/training/handouts/excelgrade/

http://www.utexas.edu/its/training/handouts/excelgrade/

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