Calculations

IV. BASIC STATISTICS CALCULATIONS CENTRAL TENDENCY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 29 Calculations Calculations are presented in the following topic areas: C Measures of Central Tendency C Student's t C Measures of Dispersion C Statistical Inference C Probability C Confidence Intervals C Z Values C ANOVA Measures of Central Tendency Measures of central tendency represent different ways of characterizing the central
value of a collection of data. Three of these measures will be addressed here: mean,
mode and median. The Mean (X-bar, X 6 ) The mean is the sum total of all data values divided by the number of data points. Example 4.6: (9 Numbers) 5 3 7 9 8 5 4 5 8 Find X 6: Answer: 6 The arithmetic mean is the most widely used measure of central tendency. Advantages of using the mean: C It is the center of gravity of the data C It uses all data C No sorting is needed IV. BASIC STATISTICS CALCULATIONS CENTRAL TENDENCY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 30 Measures of Central Tendency (Continued) The Mean (X-bar, X 6 ) (Continued) Disadvantages of using the mean: C Extreme data values may distort the picture C It can be time consuming C The mean may not be the actual value of any data points The Mode The mode is the most frequently occurring number in a data set. Example 4.7: (9 Numbers) 5 3 7 9 8 5 4 5 8 Find the mode: Answer: 5 Note: It is possible for groups of data to have more than one mode. Advantages of using the mode: C No calculations or sorting is necessary C It is not influenced by extreme values C It is an actual value C It can be detected visually in distribution plots Disadvantage of using the mode: C The data may not have a mode IV. BASIC STATISTICS CALCULATIONS CENTRAL TENDENCY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 31 Measures of Central Tendency (Continued) The Median (Midpoint) The median is the middle value when the data is arranged in ascending or
descending order. For an even set of data, the median is the average of the middle
two values. Examples 4.8: (10 Numbers) 2 2 2 3 4 6 7 7 8 9 (9 Numbers) 2 2 3 4 5 7 8 8 9 Find the median: Answer: 5 for both examples Advantages of using the median: C Provides an idea where most data are located C Little calculation required C Insensitivity to extreme values Disadvantages of using the median: C The data must be sorted and arranged C Extreme values may be important C Two medians cannot be averaged to obtain a combined median C The median will have more variation (between samples) than the average (X 6) For a Normal Distribution For a Right Skewed Distribution Figure 4.6 A Comparison of Central Tendency in Normal and Skewed Distribution IV. BASIC STATISTICS CALCULATIONS MEASURES OF DISPERSION CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 32 Measures of Dispersion Other than central tendency, the other important parameter to describe a set of data
is spread or dispersion. Three main measures of dispersion will be reviewed: range,
variance, and standard deviation. Range (R) The range of a set of data is the difference between the largest and smallest values. Example 4.9: (9 Numbers) 5 3 7 9 8 5 4 5 8 Find R: Answer: 6 Variance ( F 2 , S 2 ) The variance, F 2 or S 2 , equal to the sum of the squared deviations from the mean, divided by the sample size. The formulas for variance are: The variance is equal to the standard deviation squared. Standard Deviation ( F , s) The standard deviation is the square root of the variance. Note: N is used for a population, and n - 1 for a sample (to remove potential bias in relatively small samples - less than 30) Coefficient of Variation (COV) The coefficient of variation equals the standard deviation divided by the mean and
is expressed as a percentage. IV. BASIC STATISTICS CALCULATIONS MEASURES OF DISPERSION CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 33 The Classic Method of Calculating Standard Deviation Calculate the standard deviation of the following data set using the formula: Arrange the data into a table as follows: SAMPLE 1
2
3
4
5
6
7
8
9 10
11
12
13
14
15 X 162
176
160
142
125
159
145
167
114
120
119
180
154
125
142 X 6 146
146
146
146
146
146
146
146
146
146
146
146
146
146
146 (X-X 6) +16
+30
+14
- 4 -21 +13
- 1
+21 -32
-26
-27 +34 + 8
-21 - 4 (X-X 6) 2 256
900
196 16 441
169 1 441 1024 676
729 1156 64 441 16 3X = 2190 3(X-X 6) 2 = 6526 Calculate the average: C Compute the deviations (X-X 6) and (X-X6) 2 C Sum the squares of the deviations 3(X-X 6) 2 C Calculate the standard deviation: S is the standard deviation of the sample (21.6), which is used as an estimate for
the population from which the sample was taken. IV. BASIC STATISTICS CALCULATIONS MEASURES OF DISPERSION CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 34 Shortcut Formulas for Standard Deviation These formulas yield the same results as shown on the prior page. The two
formulas are algebraically the same. They are called “shortcut” because they are
convenient for use with some computers and calculators for messy data. Consider
the following example: X 6.6
6.8
7.3
7.4
7.7 'X = 35.8 X 2 43.56
46.24
53.29
54.76
59.29 'X 2 = 257.14 Determine X 6 and s using a Calculator Formerly this Primer attempted to instruct the student on how to determine X 6 and standard deviation on a Sharp calculator. However, many varieties of calculators
can accomplish this task. The functions on all of these calculators are subject to
change. It should be recognized that most technical people determine the mean and
dispersion using a calculator. The following general procedures apply: 1. Turn on the calculator and put it in statistical mode 2. Enter all observation values 3. Determine the sample mean (X 6) 4. Determine the population standard deviation F or sample standard deviation(S) Other Ways to get Standard Deviation Standard deviation can be determined using probability paper. Since the advent of
computer programs this is rarely done. Standard deviation can be estimated from
control charts using R 6. This technique is discussed at some length in Section V of this Primer and is tied in to the determination of process capability. IV. BASIC STATISTICS CALCULATIONS MEASURES OF DISPERSION CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 35 The Central Limit Theorem If a random variable X has mean µ and finite variance F 2 , as n increases, X 6 approaches a normal distribution with mean µ and variance . Where, and n is the number of observations on which each mean is based. Figure 4.7 Graphical Illustration of the Central Limit Theorem The Central Limit Theorem States: C The sample means (X 6s) will be more normally distributed around : than individual readings (Xs). The distribution of sample means approaches
normal regardless of the shape of the parent population. This is why X 6 - R control charts work! C The spread in sample means (X 6s) is less than Xs with the standard deviation of X 6's equal to the standard deviation of the population (individuals) divided by the square root of the sample size. S X 6 is referred to as the standard error of the mean: Example 4.10: Assume the following are weight variation results: X 6 = 20 grams and F = 0.124 grams. Estimate S X 6 for a sample size of 4: Solution: IV. BASIC STATISTICS CALCULATIONS PROBABILITY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 36 There is always a 100 % probability that a piece of toast will
land buttered side down on new carpet. From “Murphy's Laws” Probability Most inspection and quality control theory deals with statistics to make inferences
about a population based on information contained in samples. The mechanism we
use to make these inferences is probability. Conditions for Probability The probability of any event (E) lies between 0 and 1. The sum of the probabilities
of all possible events (E) in a sample space (S) = 1. Simple Events An event that cannot be decomposed is a simple event (E). The set of all sample
points for an experiment is called the sample space (S). If an experiment is repeated a large number of times, (N) and the event (E) is
observed n E times, the probability of E is approximately: Example 4.11: The probability of observing 3 on the toss of a single die is: Example 4.12: What is the probability of getting 1, 2, 3, 4, 5, or 6 by throwing a die? IV. BASIC STATISTICS CALCULATIONS PROBABILITY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 37 Compound Events Compound events are formed by a composition of two or more events. They consist of more than one point in the sample space. For example, if two dice are tossed, what is the probability of getting an 8? A die and a coin are tossed. What is the probability of getting a 4 and tail? The two most important probability theorems are the additive and multiplicative (covered later in this Section). For the following discussion, E A = A and E B = B. I. Composition. Consists of two possibilities -- a union or intersection. A. Union of A and B. If A and B are two events in a sample space (S), the union of A and B (A c B) contains all sample points in event A or B or both. Example 4.13: In the die toss of Example 4.12 consider the following: If A = E 1 , E 2 and E 3 (numbers less than 4) and B = E 1 , E 3 and E 5 (odd numbers), then A c B = E 1 , E 2 , E 3 and E 5 . B. Intersection of A and B. If A and B are two events in a sample space (S), the intersection of A and B (A 1 B) is composed of all sample points that are in both A and B. Example 4.14: Refer to Example 4.13. A 1 B = E 1 and E 3 Figure 4.8 Venn Diagrams Illustrating Union and Intersection IV. BASIC STATISTICS CALCULATIONS PROBABILITY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 38 Compound Events (Continued) II. Event Relationships. There are three relationships in finding the probability of an event: complementary, conditional and mutually exclusive. A. Complement of an Event The complement of an event A is all sample points in the sample space (S),
but not in A. The complement of A is 1-P A . Example 4.15: If P A (cloudy days) is 0.3, the complement of A would be 1 - P A = 0.7 (clear). B. Conditional Probabilities The conditional probability of event A given that B has occurred is: Example 4.16: If the event A (rain) = 0.2, and the event B (cloudiness) = 0.3, what is
the probability of rain on a cloudy day? (Note that it will not rain without clouds) Two events A and B are said to be
independent if either: P(A|B) = P(A) or P(B|A) = P(B) However,
P(A|B) = 0.67 and P(A) = 0.2= no equality, and
P(B|A) = 1.00 and P(B) = 0.3 = no equality Therefore, the events are said to be dependent. IV. BASIC STATISTICS CALCULATIONS PROBABILITY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 39 Compound Events (Continued) C. Mutually Exclusive Events If event A contains no sample points in common with event B, then they are
said to be mutually exclusive. Example 4.17: Obtaining a 3 or a 2 on the toss of a single die are mutually exclusive
events. The probability of observing both events simultaneously is zero. The
probability of obtaining either a 3 or a 2 is: D. Testing for Event Relationships Example 4.18: Refer to Example 4.13. Event A: E 1 , E 2 , E 3 Event B: E 1 , E 3 , E 5 Are A and B, mutually exclusive, complementary, independent or dependent? A and
B contain two sample points in common so they are not mutually exclusive. They
are not complementary because B does not contain all points in S that are not in A. To determine if they are independent requires a check. By definition, events A and B are dependent. IV. BASIC STATISTICS CALCULATIONS PROBABILITY CQT 2003 © QUALITY COUNCIL OF INDIANA IV - 40 The Additive Law If the two events are not mutually exclusive: 1. P (A c B) = P(A) + P(B) - P (A 1 B) Note that P (A c B) is shown in many texts as P (A + B) and is read as the probability of A or B. Example 4.19: If one owns two cars and the probability of each car starting on a cold
morning is 0.7, what is the probability of getting to work? P (A c B) = 0.7 + 0.7 - (0.7 x 0.7)
= 1.4 - 0.49
= 0.91 = 91 % If the two events are mutually exclusive, the law reduces to: 2. P (A c B) = P(A) + P(B) also P (A + B) = P(A) + P(B) Example 4.20: If the probability of finding a black sock in a dark room is 0.4 and the
probability of finding a blue sock is 0.3, what is the chance of finding a blue or black
sock? P (A c B) = 0.4 + 0.3 = 0.7 = 70 % Note: The problem statement centers around the word “or.” Will car A or B start?
Will I get a black sock or blue sock?

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