Chap 2 Introduction to Statistics

This chapter gives overview of statistics including histogram construction, measures of central tendency, and dispersion

INTRODUCTION TO STATISTICS Statistics – deriving relevant information

from data Deals with

Collection of data – census, GDP, football, accident, no. of employees (male, female , department, etc)

Collection , tabulation, analysis, interpretation, an presentation of quantitative data – can make some conclusions on sample or population studied, make decisions on quality

INTRODUCTION TO STATISTICS

Use of statistics in quality deals with second meaning. – inductive statistics

Examples : What can we learn from the data? What conclusions can be drawn? What does the data tell about our process

and product performance? etc.

INTRODUCTION TO STATISTICS Understand the use of statistics vital

in business to make decisions based on facts in conducting business improvements in controlling and monitoring process,

products or service performance Application of statistics to real life

problems such as for quality problems will result in improved organizational performance

Collection of data

Collect Data – direct observation or indirect through written or verbal questions (market research, opinion polls)

Direct observation measured, visual checking, classified as variables and attributes

Variables data – measurable quality characteristics

Attributes – characteristics not measured but classified as conforming or non-conforming

Collection of data

Data collected with purpose Find out process conditions For improvement

Variables – quality characteristics that are measurable and countable CONTINUOUS - Dimensions, weight, height,

etc. (meter, gallon, p.s.i., etc.) DISCRETE - numbers that exhibit gaps,

countable, (no. of defective parts, no. of defects/car, Whole numbers, 1, 2, 3….100)

Collection of data Attributes - quality characteristics that are non-

measurable and ‘those we do not want to measure’

Example : surface appearance, color, Acceptable, non-acceptable conforming, non-conf.

Data collected in form of discrete values Variables (weight of sugar) CAN be classified as

attributes  weight within limits – number of

conforming outside limits – no. of non conforming

0 1 3 0 1 0 1 0

1 5 4 1 2 1 2 0

1 0 2 0 0 2 0 1

2 1 1 1 2 1 1

0 4 1 3 1 1 1

1 3 4 0 0 0 0

1 3 0 1 2 2 3

Summarizing Data Consider this data set on number of Daily Billing errors

Data in this from MeaninglessNot effectiveDifficult to use

Need to summarize data in the form of: Graphical – Freq. Dist., Histogram, Graphs,

Charts, Diagrams Analytical – Measures of central tendency,

Measure of dispersion

Frequency Distribution (FD)

Summary of how data (observations) occur within each subdivision or groups of observed values

Help visualize distribution of data Can see how total frequency is distributed Two types : Ungrouped data – listing of observed values Grouped data – lump together observed values

23.522.5

22.521.5

21.520.5

FD - Ungrouped Data

1. Establish array, arrange in ascending or descend (as in column 1)

2. Tabulate the frequency – place tally marking in column 2

3. Present in graphical form – Histogram, Relative freq. distr.

No of errors Tally mark Frequency

0 /////////// 13

1 ////

2 /////

3 ////

4

5

FD – Ungrouped data

No error Freq Relative freq

Cumulative freq

Rel cum freq

0 15 0.29 15 0.29

1 20 0.38 35 0.67

2 8 0.15 43 0.83

3 5 0.10 48 0.92

4 3 0.06 51 0.98

5 1 0.02 52 1.00

Total 52

4 graphical representations

1. Frequency histogram

2. Relative freq histogram

3. Cumulative frequency histogram

4. Relative cum frequency histogram 0 1 2 3 4 5

1412108642

Frequency

Frequency Distribution For Grouped Data Data which are continuous variable need grouping

Steps1. Collect data and construct tally sheet Make tally - coded if necessary Too many data – group into cells Simplify presentation of distribution Too many cells – distort true picture Too few cells – too concentrated No of cells – judgment by analyst – trial and error Generally 5-20 cells Less than 100 data – use 5 –9 cells 100 – 500 data – use 8 to 17 cells More than 500 – use 15 to 20 cells

Midpoint

UPPER BOUNDARY

CELL

CELL NOMENCLATURE

Cell interval (i)

2. Determine the range

R = XH - XL R = range XH = highest value of data XL = lowest value of data Example : If highest number is 2.575 and lowest number

is 2.531, then R = XH - XL = 2.575 – 2.531 = 0.044

3. Determine the cell interval Cell interval = distance between adjacent cell midpoints.

If possible, use odd interval values e.g. 0.001, 0.07, 0.5 , 3; so that midpoint values will have same no. decimal places as data values.

Use Sturgis rule. i = R/(1+ 3.322 log n) Trial and error h = R/i ;h= number of cells or cllases Assume i = 0.003; h = 0.044/0.003 = 15 cells Assume i = 0.005; h = 0.044/0.005 = 9 cells Assume ii = 0.007; h = 0.044/0/.007 = 6 cells Cell interval 0.005 with 9 cells will give best presentation

of data. Use guidelines in step 1.

4. Determine cell midpoints

MPL = XL + i/2 (do not round) = 2.531 + 0.005/2 = 2.533 1st cell have 5 different values (also the other

cells)

2.531 2.532 2.533 2.534 2.535

2.533

2.538

5. Determine cell boundaries

Limit values of cell lower upper To avoid ambiguity in putting data Boundary values have an extra decimal

place or sig. figure in accuracy that observed values

+ 0.0005 to highest value in cell - 0.0005 to lowest value in cell

6. Tabulate cell frequency Post amount of numbers in each cell Frequency distribution table

Cell boundary Cell MP Freq.

2.531 – 2.535 2.533 6

2.536 – 2.540 2.538 8

2.541 – 2.545 2.543 12

2.546 – 2.550 2.553 13

2.551 – 2.555 2.553 20

2.556 – 2.560 2.563 19

2.561 – 2.565 2.563 13

2.566 – 2.570 2.568 11

2.571 – 2.575 2.573 8

110

Freq dist gives better view of central value and how data dispersed than the unorganized data sheet

Histogram – describes variation in process Used to solve problems determine process capability compare with specifications suggest shape of distribution indicate data discrepancies, e.g. gaps

Characteristics Of Frequency Distribution Symmetry, Number of modes (one, two or

multiple), Peakedness of data

Sym.

SkewRight

SkewLeft

Bi-modal

‘very peak’leptokurtic

flatterplatykurtic

Characteristics of Frequency Distribution

F.D. can give sufficient info to provide basis for decision making.

Distributions are compared regarding:-

Location Spread Shape

Descriptive Statistics Analytical method allow comparison between

data 2 main analytical methods for describing data

Measures of central tendency Measures of dispersion

Measures of central tendency of a distribution - a numerical value that describes the central position of data

3 common measures mean median mode

Measure of Central Tendency Mean - most common measure used What is middle value? What is average

number of rejects, errors, dimension of product?

Mean for Ungrouped Data - unarranged x (x bar)

n

xxx

n

xX n21

n

1ii

Mean

ExampleA QA engineer inspects 5 pieces of a tyre’s thread depth (mm). What is the mean thread depth?

x1 = 12.3 x2 = 12.5 X3 = 12.0.x4 = 13.0 x5 = 12.8

mm12.55

62.5

5

Σxx i

Mean - Grouped Data

When data already grouped in frequency distribution

fi (n)= sum. of freq.

fi = freq in the ith cell

n = no. of cells/classxi = mid point in ith cell

i

h

1iii

Σf

xfx

Mean - Grouped Data

Cell (i) Class boundary

Mid Point (xi)

Freq (fi)

Fixi fi fixi

1 1 – 20 10 2 20 2

2 21 – 40 30 10 300 12

3 41 - 60 50 20 1000 32

4 61 – 80 70 12 840 44

5 81 -100 90 6 540 50

Totals 2700

i

ii

fxf

x

= 2700/50 = 54

Weighted average

Tensile tests aluminium alloy conducted with differentnumber of samples each time. Results are as follows: 1st test : x1 = 207 MPa n = 5

2nd test : x2 = 203 MPa n = 6

3rd test : x3 = 206 MPa n = 3

or use sum of weights equals 1.00 W1 = 5/(5+6+3) = 0.36W2 = 6/(5+6+3) = 0.43W3 = 3/(5+6+3) = 0.21 Total = 1.00

n

1ii

n

1iii

w

w

xwx

MPa205365

(206)3(203)6(5)(207)xw

xw = weighted avg.

wi = weight of ith average

Median – Ungrouped Data Median – value of data which divides total

observation into 2 equal parts Ungrouped data – 2 possibilities When total number of data (N) is a) odd or b)

even If N is odd ; (N+1/2)th value is median eg. 3 4 5 6 8 N+1/2=6/2=3 ,

3rd no. If N is even eg. 3 5 7 9 ½ of (5+7)=6

NOTE: ORDER THE NUMBERS FIRST!

Median – Grouped Data Need to find cell / class having middle value &

interpolating in the cell using

Lm = lower boundary of cell with the medianCfm = Cum. freq. of all cells below Lmfm =class/cell freq. where median occursi = cell interval

ExampleMD = 40.5 + 10

= 53.5

if

cf2n

Lxm

m

m0.5

Measures of dispersion

describes how the data are spread out or scattered on each side of central value

both measures of central tendency & dispersion needed to describe data

Exams Results Class 1 – avg. : 60.0 marks highest : 95 lowest : 25 Class 2 – avg. : 60.0 marks highest : 100 lowest : 15 marks

Measures of dispersion

Main types – range, standard deviation, and variance

Range – difference bet. highest & lowest value

R = XH - XL Standard deviation Variance – standard deviation squared Large value shows greater variability or

spread

Standard deviation

For Ungrouped Data s = sample std. dev.

xi = observed value

x = average n = no. of observed value

or use

1n

xxs

n

1i

2i

1nn

xxn

s

2

i

n

1i

2i

n

1i

Standard deviation – grouped data

Cell (i) Class boundary

Mid Point (xi)

Freq (fi)

Fixi fi fixi

1 1 – 20 10 2 20 2

2 21 – 40 30 10 300 12

3 41 - 60 50 20 1000 32

4 61 – 80 70 12 840 44

5 81 -100 90 6 540 50

Totals 2700

20.6424.494950

(2700)(166,600)50 2

1)(nn

xfxfn

s

h

1

h

1

2ii

2ii

NOTE: DO NOT ROUND OFF fixi & fixi2ACCURACY AFFECTED

Concept Of Population and Sample Total daily prod. of steel shaft. Year’s Prod. Volume of calculators Compute x and s sample statistics True Population Parameters and Why sample? not possible measure population costs involved 100% manual inspection –

accuracy/error

Population

Sample

Concept Of Population and Sample

SAMPLEStatistics, x , s

POPN.Parameter - mean - std. dev.

Normal Distribution

Also called Gaussian distribution Symmetrical, unimodal, bell-shaped dist

with mean, median, mode same value Popn. curve – as sample size cell interval

- get smooth polygon

ND

Normal Distribution

Much of variation in nature & industry follow N.D.

Variation in height of humans, weight of elephants, casting weights, size piston ring

Electrical properties, material – tensile strength, etc.

Example - ND

Characteristics of ND

Can have different mean but same standard deviation

Different standard deviation but same mean

Relationship between std deviation and area under curve

Normal Distribution Example

Need estimates of mean and standard deviation and the Normal Table

Example : From past experience a manufacturer

concludes that the burnout time of a particular light bulb follows a normal distribution. Sample has been tested and the average (x ) found to be 60 days with a standard deviation () of 20 days. How many bulbs can be expected to be still working after 100 days.

Solution Problem is actually to find area under the curve beyond 100 days Sketch Normal distribution and shade the area needed Calculate z value corresponding to x value using formula Z=(xi - )/ = (100-60)/20 = +2.00 Look in the Normal Table for z = +2.00 – gives area under curve as

0.9773 But, we want x >100 or z > 2.00. Therefore Area = 1.000 – 0.9773

= 0.0227, i.e. 2.27% probability that life of light bulb is > 100 hours

μ = 60

σ =20

x0 100

Test For Normality To determine whether data is normal Probability Plot - plot data on normal probability

paper Steps1. Order the data2. Rank the observations3. Calculate the plotting position

i= rank , n=sample size, PP= plotting position in %

4. Label data scale5. Plot the points on normal probability paper6. Attempt to fit by eye ‘best line’7. Determine normality

n

)5.0i(100PP

Example