1. Chap 2 Introduction to Statistics
This chapter gives overview of statistics including histogram construction, measures of central tendency, and dispersion

2. 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

3. 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.

4. 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

5. 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

6. 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)

7. 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

8. Summarizing Data
Consider this data set on number of Daily Billing errors
Data in this from
Meaningless
Not effective
Difficult to use 1 3 5 4 2

9. Need to summarize data in the form of:
Graphical – Freq. Dist., Histogram, Graphs, Charts, Diagrams
Analytical – Measures of central tendency, Measure of dispersion

10. 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

11. FD - Ungrouped Data
Establish array, arrange in ascending or descend (as in column 1)
Tabulate the frequency – place tally marking in column 2
Present in graphical form – Histogram, Relative freq. distr.
No of errors
Tally mark
Frequency
/////////// 13 1
//// 2
///// 3 4 5

12. FD – Ungrouped data No error Freq Relative freq Cumulative freq1 2 3 4 5 14 12 10 8 6
4 graphical representations
Frequency histogram
Relative freq histogram
Cumulative frequency histogram
Relative cum frequency histogram
Frequency
No error
Freq
Relative freq
Cumulative freq
Rel cum freq 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
0.06 51
0.98
0.02 52
1.00
Total

13. Frequency Distribution For Grouped Data
Data which are continuous variable need grouping
Steps
1. 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

14. Midpoint
UPPER BOUNDARY
CELL
CELL NOMENCLATURE
Cell interval (i)

15. 2. Determine the range R = XH - XL R = range
XH = highest value of data
XL = lowest value of data
Example :
If highest number is and lowest number is 2.531, then
= – 2.531
= 0.044

16. 3. Determine the cell interval
Cell interval = distance between adjacent cell midpoints. If possible, use odd interval values e.g , 0.07, 0.5 , 3; so that midpoint values will have same no. decimal places as data values.
Use Sturgis rule.
i = R/( 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/ = 9 cells
Assume ii = 0.007; h = 0.044/0/.007 = 6 cells
Cell interval with 9 cells will give best presentation of data. Use guidelines in step 1.

17. 4. Determine cell midpoints
MPL = XL + i/2 (do not round)
= /2 = 2.533
1st cell have 5 different values (also the other cells)
2.533
2.538

18. 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
to highest value in cell
to lowest value in cell

19. 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 20
2.556 – 2.560
2.563 19
2.561 – 2.565
2.566 – 2.570
2.568 11
2.571 – 2.575
2.573
110

20. 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

21. Characteristics Of Frequency Distribution
Symmetry, Number of modes (one, two or multiple), Peakedness of data
Bi-modal
Sym.
Skew
Right
Left
flatter
platykurtic
‘very peak’
leptokurtic

22. Characteristics of Frequency Distribution
F.D. can give sufficient info to provide basis for decision making.
Distributions are compared regarding:-
Shape
Spread
Location

23. 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

24. 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)

25. Mean
Example
A 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

26. 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/class
xi = mid point in ith cell

27. Mean - Grouped Data = 2700/50 = 54 Cell (i) Class boundary
Mid Point (xi)
Freq (fi)
Fixi
fi
fixi 1
1 – 20 10 2 20
21 – 40 30
300 12 3 50
1000 32 4
61 – 80 70
840 44 5 90 6
540
Totals
2700
= 2700/50 = 54

28. Weighted average xw = weighted avg.
Tensile tests aluminium alloy conducted with different
number 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) = 0.36
W2 = 6/(5+6+3) = 0.43
W3 = 3/(5+6+3) = Total = 1.00
xw = weighted avg.
wi = weight of ith average

29. 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 N+1/2=6/2=3 , 3rd no.
If N is even
eg ½ of (5+7)=6
NOTE: ORDER THE NUMBERS FIRST!

30. Median – Grouped Data
Need to find cell / class having middle value & interpolating in the cell using
Lm = lower boundary of cell with the median
Cfm = Cum. freq. of all cells below Lm
fm =class/cell freq. where median occurs
i = cell interval
Example
MD =
= 53.5

31. 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

32. 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

33. Standard deviation For Ungrouped Data s = sample std. dev.
xi = observed value
x = average
n = no. of observed value
or use

34. Standard deviation – grouped data
Cell (i)
Class boundary
Mid Point (xi)
Freq (fi)
Fixi
fi
fixi 1
1 – 20 10 2 20
21 – 40 30
300 12 3 50
1000 32 4
61 – 80 70
840 44 5 90 6
540
Totals
2700
NOTE: DO NOT ROUND OFF fixi & fixi2
ACCURACY AFFECTED

35. 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

36. Concept Of Population and Sample
Statistics, x , s
POPN.
Parameter
 - mean
 - std. dev.

37. 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

38. 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.

39. Example - ND

40. Characteristics of ND
Can have different mean but same standard deviation

41. Different standard deviation but same mean

42. Relationship between std deviation and area under curve

43. 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.

44. 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 = – gives area under curve as
But, we want x >100 or z > Therefore Area = – = , i.e. 2.27% probability that life of light bulb is > 100 hours
σ =20
μ = 60
100 x

45. Test For Normality To determine whether data is normal
Probability Plot - plot data on normal probability paper
Steps
Order the data
Rank the observations
Calculate the plotting position
i= rank , n=sample size,
PP= plotting position in %
Label data scale
Plot the points on normal probability paper
Attempt to fit by eye ‘best line’
Determine normality

46. Example