View
222
Download
0
Category
Preview:
Citation preview
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 1/87
©The McGraw-Hill Companies, Inc. 2008 McGraw-Hill/Irwin
What is Statistics
Chapter 1
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 2/87
2
GOALS
Understand why we study statistics.
Explain what is meant by descriptivestatistics and inf erential statistics.
Distinguish between a qualitative variableand a quantitative variable.
Describe how a discrete variable is diff erentf rom a continuous variable.
Distinguish among the nominal, or dinal,interval, and ratio levels of measurement.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 3/87
3
What is Meant by Statistics?
Statistics is the science of
collecting, or ganizing, presenting,analyzing, and interpreting numerical data to assist in making
more eff ective decisions.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 4/87
4
Who Uses Statistics?
Statistical techniques are used
extensively by marketing,accounting, quality control,consumers, prof essional sportspeople, hospital administrators,
educators, politicians, physicians,etc...
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 5/87
5
Types of Statistics ± DescriptiveStatistics
Descriptive Statistics - methods of or ganizing,summarizing, and presenting data in an inf ormative way.
EXAMPLE 1: A Gallup poll f ound that 49% of the people in a survey knew the name of the f irst book of the Bible. The statistic 49 describes the number out of every 100persons who knew the answer.
EXAMPLE 2: Accor ding to Consumer Reports, General Electric washing machineowners reported 9 problems per 100 machines during 2001. The statistic 9describes the number of problems out of every 100 machines.
Inf erential Statistics: A decision, estimate,prediction, or generalization about apopulation, based on a sample.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 6/87
6
Population versus Sample
A population is a collection of all possible individuals, objects, or measurements of interest.
A sample is a portion, or part, of the population of interest
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 7/87
7
Types of Variables
A. Qualitative or Attribute variable - thecharacteristic being studied is nonnumeric.
EXAMPLES: Gender, religious aff iliation, type of automobileowned, state of birth, eye color are examples.
B. Quantitative variable - inf ormation is reported
numerically.EXAMPLES: balance in your checking account, minutes
remaining in class, or number of children in a f amily.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 8/87
8
Quantitative Variables - Classifications
Quantitative variables can be classif ied as either discreteor continuous.
A. Discrete variables: can only assume certain valuesand there are usually ³gaps´ between values.
EXAMPLE: the number of bedrooms in a house, or the number of hammers sold at the localHome Depot (1,2,3,«,etc).
B. Continuous variable can assume any value within aspecif ied range.
EXAMPLE: The pressure in a tire, the weight of a pork chop, or the height of students in aclass.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 9/87
9
Summary of Types of Variables
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 10/87
10
Four Levels of Measurement
Nominal level - data that isclassif ied into categories and cannot be arranged in anyparticular or der.
EXAMPLES: eye color, gender,
religious aff iliation.
Or dinal level ± involves dataarranged in some or der, but thediff erences between datavalues cannot be determined or
are meaningless.EXAMPLE: During a taste test of
4 sof t drinks, Mellow Yellowwas ranked number 1, Spritenumber 2, Seven-up number 3, and Orange Crush number 4.
Interval level - similar to the or dinallevel, with the additionalproperty that meaningf ulamounts of diff erences between data values can be determined.
There is no natural zero point.EXAMPLE: Temperature on the
Fahrenheit scale.
Ratio level - the interval level withan inherent zero starting point.Diff erences and ratios are
meaningf ul f or this level of measurement.EXAMPLES: Monthly incomeof sur geons, or distancetraveled by manuf acturer¶srepresentatives per month.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 11/87
11
Summary of the Characteristics for Levels of Measurement
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 12/87
©The McGraw-Hill Companies, Inc. 2008 McGraw-Hill/Irwin
Describing Data:Frequency Tables, FrequencyDistributions, and Graphic Presentation
Chapter 2
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 13/87
13
GOALS
Or ganize qualitative data into a f requencytable.
Present a f requency table as a bar chart or apie chart.Or ganize quantitative data into a f requencydistribution.Present a f requency distribution f or quantitative data using histograms, f requencypolygons, and cumulative f requencypolygons.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 14/87
14
Bar Charts
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 15/87
15
Pie Charts
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 16/87
16
Pie Chart Using Excel
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 17/87
17
Frequency Distribution
A Frequencydistribution is a
grouping of data intomutually exclusivecategories showing the number of
observations in eachclass.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 18/87
18
Frequency Table
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 19/87
19
Relative Class Frequencies
Class f requencies can be converted torelative class frequencies to show the
f raction of the total number of observations in each class.
A relative f requency captures the relationshipbetween a class total and the total number of
observations.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 20/87
20
Frequency Distribution
Class midpoint: A point that divides a classinto two equal parts. This is the average
of the upper and lower class limits.Class f requency: The number of observations in each class.
Class interval: The class interval is
obtained by subtracting the lower limit of a class f rom the lower limit of the nextclass.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 21/87
21
EXAMPLE ± Creating a FrequencyDistribution Table
Ms. Kathryn Ball of AutoUS Awants to develop tables,charts, and graphs to showthe typical selling price on
various dealer lots. Thetable on the right reportsonly the price of the 80vehicles sold last month atWhitner Autoplex.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 22/87
22
EXAMPLE ± Creating a FrequencyDistribution Table
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 23/87
23
Constructing a Frequency Table -Example
Step 1: Decide on the number of classes. A usef ul recipe to determine the number of classes (k ) isthe ³2 to the k rule.´ such that 2k > n.There were 80 vehicles sold. So n = 80. If we try k = 6, which
means we would use 6 classes, then 26 = 64, somewhat lessthan 80. Hence, 6 is not enough classes. If we let k = 7, then 27
128, which is greater than 80. So the recommended number of classes is 7.
Step 2: Determine the class interval or width.The f ormula is: i u (H-L)/ k where i is the class interval, H isthe highest observed value, L is the lowest observed value,and k is the number of classes.($35,925 - $15,546)/7 = $2,911Round up to some convenient number, such as a multiple of 10
or 100. Use a class width of $3,000
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 24/87
24
Step 3: Set the individual class limits
Constructing a Frequency Table -Example
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 25/87
25
Step 4: Tally the vehicleselling prices into the
classes.
Step 5: Count the number of items in each class.
Constructing a Frequency Table
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 26/87
26
Relative Frequency Distribution
To convert a f requency distribution to a rel ativ e f requencydistribution, each of the class f requencies is divided by thetotal number of observations.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 27/87
27
Graphic Presentation of aFrequency Distribution
The three commonly used graphic f orms
are:Histograms
Frequency polygons
Cumulative f requency distributions
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 28/87
28
Histogram
Histogram f or a f requency distribution based on quantitative data is very similar to the bar chart showing thedistribution of qualitative data. The classes are marked on the horizontal axis and the class f requencies on the verticalaxis. The class f requencies are represented by the heightsof the bars.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 29/87
29
Histogram Using Excel
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 30/87
30
Frequency Polygon
A frequency polygonalso shows the shapeof a distribution and is
similar to a histogram.
It consists of linesegments connecting the points f ormed bythe intersections of theclass midpoints and theclass f requencies.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 31/87
31
Cumulative Frequency Distribution
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 32/87
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 33/87
©The McGraw-Hill Companies, Inc. 2008 McGraw-Hill/Irwin
Describing Data:Numerical Measures
Chapter 3
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 34/87
34
GOALS
Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean. Explain the characteristics, uses, advantages, and disadvantages of eachmeasure of location.
Identif y the position of the mean, median, and mode f or both symmetric and skewed distributions. Compute and interpret the range, mean deviation, variance, and standar d deviation. Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion. Understand Chebyshev¶s theorem and the Empirical Rule as they relate to a
set of observations.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 35/87
35
Characteristics of the Mean
The arithmetic mean is the most widely used measure of location. It requires the interval
scale. Its major characteristics are: ± All values are used.
± It is unique.
± The sum of the deviations f rom the mean is 0.
± It is calculated by summing the values and dividing by the number of values.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 36/87
36
Population Mean
For ungrouped data, the population mean is thesum of all the population values divided by the
total number of population values:
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 37/87
37
EXAMPLE ± Population Mean
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 38/87
38
Sample Mean
For ungrouped data, the sample mean is the sum of all the sample values
divided by the number of samplevalues:
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 39/87
39
EXAMPLE ± Sample Mean
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 40/87
40
Properties of the Arithmetic Mean
Every set of interval-level and ratio-level data has a mean.
All the values are included in computing the mean.
A set of data has a unique mean.
The mean is aff ected by unusually lar ge or small data values. The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value f rom the mean iszero.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 41/87
41
Weighted Mean
The weighted mean of a set of numbers X 1, X 2, ..., X n, with corresponding weights w 1,w 2, ...,w n, is computed f rom the f ollowing f ormula:
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 42/87
42
EXAMPLE ± Weighted Mean
The Carter Construction Company pays its hourly employees $16.50,$19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the$25.00 rate. What is the mean hourly rate paid the 26 employees?
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 43/87
43
The Median
The Median is the midpoint of the valuesaf ter they have been or dered f rom the
smallest to the lar gest. ± There are as many values above the median as
below it in the data array.
± For an even set of values, the median will be the
arithmetic average of the two middle numbers.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 44/87
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 45/87
45
EXAMPLES - Median
The ages f or a sampleof f ive collegestudents are:
21, 25, 19, 20, 22
Arranging the data in
ascending or der gives:
19, 20, 21, 22, 25.
The heights of f our basketballplayers, in inches, are:
76, 73, 80, 75
Arranging the data in ascending or der gives:
73, 75, 76, 80.
Thus the median is 75.5
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 46/87
46
The Mode
The mode is the value of the observation that appears most f requently.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 47/87
47
Example - Mode
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 48/87
48
Mean, Median, Mode Using Excel
Table 2±4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price. The mean and the median selling prices are reported in the f ollowing Excel output. There are 80 vehicles in the study. So thecalculations with a calculator would be tedious and prone to error.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 49/87
49
Mean, Median, Mode Using Excel
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 50/87
50
The Relative Positions of the Mean,Median and the Mode
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 51/87
51
The Geometric Mean
Usef ul in f inding the average change of percentages, ratios, indexes,or growth rates over time.
It has a wide application in business and economics because we areof ten interested in f inding the percentage changes in sales, salaries,
or economic f igures, such as the GDP, which compound or build on each other.
The geometric mean will always be less than or equal to thearithmetic mean.
The geometric mean of a set of n positive numbers is def ined as thenth root of the product of n values.
The f ormula f or the geometric mean is written:
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 52/87
52
EXAMPLE ± Geometric Mean
Suppose you receive a 5 percent increase in salary this year and a 15 percent increase
next year. The average annual percentincrease is 9.886, not 10.0. Why is this so?We begin by calculating the geometric mean.
098861151051 . ). )( .( GM !!
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 53/87
53
EXAMPLE ± Geometric Mean (2)
The retur n on investment ear ned by Atkinsconstruction Company f or f our successive
years was: 30 percent, 20 percent,-40 percent,and 200 percent. What is the geometric mean rate of retur n on investment?
.. ). )( . )( . )( .( 2941808203602131 44 !!!
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 54/87
54
Dispersion
Why Study Dispersion?
± A measure of location, such as the mean or themedian, only describes the center of the data. It
is valuable f rom that standpoint, but it does nottell us anything about the spread of the data.
± For example, if your nature guide told you that theriver ahead averaged 3 f eet in depth, would youwant to wade across on f oot without additionalinf ormation? Probably not. You would want toknow something about the variation in the depth.
± A second reason f or studying the dispersion in aset of data is to compare the spread in two or
more distributions.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 55/87
55
Samples of Dispersions
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 56/87
56
Measures of Dispersion
Range
Mean Deviation
Variance and Standar d Deviation
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 57/87
57
EXAMPLE ± Range
The number of cappuccinos sold at the Starbucks location in theOrange Country Airport between 4 and 7 p.m. f or a sample of 5days last year were 20, 40, 50, 60, and 80. Determine the mean deviation f or the number of cappuccinos sold.
Range = Lar gest ± Smallest value= 80 ± 20 = 60
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 58/87
58
EXAMPLE ± Mean Deviation
The number of cappuccinos sold at the Starbucks location in theOrange Country Airport between 4 and 7 p.m. f or a sample of 5days last year were 20, 40, 50, 60, and 80. Determine the mean deviation f or the number of cappuccinos sold.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 59/87
59
EXAMPLE ± Variance and StandardDeviation
The number of traff ic citations issued during the last f ive months in Beauf ort County, South Carolina, is 38, 26, 13, 41, and 22. Whatis the population variance?
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 60/87
60
EXAMPLE ± Sample Variance
The hourly wages f or a sample of part-time employees atHome Depot are:$12, $20, $16, $18,and $19. What isthe samplevariance?
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 61/87
61
Chebyshev¶s Theorem
The arithmetic mean biweekly amount contributed by the DupreePaint employees to the company¶s prof it-sharing plan is $51.54,and the standar d deviation is $7.51. At least what percent of thecontributions lie within plus 3.5 standar d deviations and minus3.5 standar d deviations of the mean?
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 62/87
62
The Empirical Rule
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 63/87
63
The Arithmetic Mean of Grouped Data
Arithmetic Mean of Grouped Data
where:
is the designation f or the sample mean
is the sum of the class f requencies
is the midpoint of each class
is the sum of the f requency of each class times the midpoint of
the class.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 64/87
64
Recall in Chapter 2, weconstructed a f requencydistribution f or the vehicleselling prices. The
inf ormation is repeated below. Determine thearithmetic mean vehicleselling price.
The Arithmetic Mean of Grouped Data -Example
Selling Price
($'000)Frequency
15 up to 18 818 up to 21 23
21 up to 24 17
24 up to 27 18
27 up to 30 8
30 up to 33 4
33 up to 36 2
80
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 65/87
65
The Arithmetic Mean of Grouped Data -Example
Selling Price
($'000)Frequency
Mid-Point
(x)fx
15 up to 18 8 16.5 132.0
18 up to 21 23 19.5 448.521 up to 24 17 22.5 382.5
24 up to 27 18 25.5 459.0
27 up to 30 8 28.5 228.0
30 up to 33 4 31.5 126.0
33 up to 36 2 34.5 69.0
80 1,845.0
The Mean Selling Priceis $23,100
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 66/87
66
Standard Deviation of Grouped Data
Variance of Grouped Data
Standard Deviation of Grouped Data
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 67/87
67
Standard Deviation of Grouped Data -Example
Ref er to the f requency distribution f or the Whitner Autoplex data used earlier. Compute the standar d deviation of the vehicle selling prices
Selling Price
($'000)Frequency
Mid-Point
(x)fx fx2
15 up to 18 8 16.5 132.0 2178.0018 up to 21 23 19.5 448.5 8745.75
21 up to 24 17 22.5 382.5 8606.25
24 up to 27 18 25.5 459.0 11704.50
27 up to 30 8 28.5 228.0 6498.00
30 up to 33 4 31.5 126.0 3969.00
33 up to 36 2 34.5 69.0 2380.50
80 1,845.0 44082.00
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 68/87
68
The standar d deviation is the most widely used measure of dispersion.
Alter native ways of describing spread of data includedetermining the locati on of values that divide a set of observations into equal parts.
These measures include quartiles, deciles, and percentiles.
Quartiles, Deciles and Percentiles
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 69/87
69
To f ormalize the computational procedure, let L p ref er to thelocation of a desired percentile. So if we wanted to f ind the 33r d percentile we would use L33 and if we wanted the median, the50th percentile, then L50.
The number of observations is n, so if we want to locate themedian, its position is at (n + 1)/2, or we could write this as
(n + 1)(P /100), where P is the desired percentile.
Percentile Computation
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 70/87
70
Percentiles - Example
Listed below are the commissions ear ned last monthby a sample of 15 brokers at Salomon SmithBar ney¶s Oakland, Calif or nia, off ice. Salomon SmithBar ney is an investment company with off ices
located throughout the United States.
$2,038 $1,758 $1,721 $1,637$2,097 $2,047 $2,205 $1,787$2,287 $1,940 $2,311 $2,054$2,406 $1,471 $1,460
Locate the median, the f irst quartile, and the thir d quartile f or the commissions ear ned.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 71/87
71
Percentiles ± Example (cont.)
Step 1: Or ganize the data f rom lowest tolar gest value
$1,460 $1,471 $1,637 $1,721
$1,758 $1,787 $1,940 $2,038
$2,047 $2,054 $2,097 $2,205
$2,287 $2,311 $2,406
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 72/87
72
Percentiles ± Example (cont.)
Step 2: Compute the f irst and thir d quartiles.Locate L25 and L75 using:
205,2$
721,1$
lyrespectivearray,in thenobservatio
12thand4ththearequartilesthirdandfirsttheTherefore,
12100
75)115( 4
100
25)115(
75
25
7525
!
!
!!!!
L
L
L L
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 73/87
73
Skewness
In Chapter 3, measures of central location f or a setof observations (the mean, median, and mode) and measures of data dispersion (e.g. range and the
standar d deviation) were introduced Another characteristic of a set of data is the shape.
There are f our shapes commonly observed: ± symmetric,
± positively skewed, ± negatively skewed,
± bimodal.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 74/87
74
Skewness - Formulas for Computing
The coeff icient of skewness can range f rom -3 up to 3.
± A value near -3, such as -2.57, indicates considerable negative skewness.
± A value such as 1.63 indicates moderate positive skewness.
± A value of 0, which will occur when the mean and median are equal,
indicates the distribution is symmetrical and that there is no skewnesspresent.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 75/87
75
Commonly Observed Shapes
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 76/87
76
Skewness ± An Example
Following are the ear nings per share f or a sample of 15 sof tware companies f or the year 2005. Theear nings per share are arranged f rom smallest tolar gest.
Compute the mean, median, and standar d deviation.Find the coeff icient of skewness using Pearson¶sestimate. What is your conclusion regar ding theshape of the distribution?
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 77/87
77
Skewness ± An Example UsingPearson¶s Coefficient
017.1
22.5$
)18.3$95.4($3)(3
22.5$115
))95.4$40.16($...)95.4$09.0($
1
95.4$
15
26.74$
222
!
!
!
!
!
7!
!!!§
s
edian X sk
n
X X s
n
X X
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 78/87
78
Skewness ± A Minitab Example
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 79/87
79
Describing Relationship between TwoVariables
One graphical technique weuse to show the relationshipbetween variables is called a
scatter diagram. To draw a scatter diagram we
need two variables. We scaleone variable along the
horizontal axis ( X
-axis) of agraph and the other variablealong the vertical axis (Y -axis).
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 80/87
80
Describing Relationship between TwoVariables ± Scatter Diagram Examples
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 81/87
81
In the Introduction to Chapter 2 we presented data f rom AutoUS A. In this case the inf ormation concer ned theprices of 80 vehicles sold last month at the Whitner
Autoplex lot in Raytown, Missouri. The data shown include the selling price of the vehicle as well as theage of the purchaser.
Is there a relationship between the selling price of a
vehicle and the age of the purchaser? Would it bereasonable to conclude that the more expensivevehicles are purchased by older buyers?
Describing Relationship between TwoVariables ± Scatter Diagram Excel Example
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 82/87
82
Describing Relationship between TwoVariables ± Scatter Diagram Excel Example
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 83/87
83
Contingency Tables
A scatter diagram requires that both of thevariables be at least interval scale.
What if we wish to study the relationshipbetween two variables when one or both arenominal or or dinal scale? In this case we tallythe results in a contingency table.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 84/87
84
Contingency Tables ± An Example
A manuf acturer of preassembled windows produced 50 windowsyester day. This mor ning the quality assurance inspector reviewed each window f or all quality aspects. Each was classif ied asacceptable or unacceptable and by the shif t on which it wasproduced. Thus we reported two variables on a single item. The
two variables are shif t and quality. The results are reported in thef ollowing table.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 85/87
85
End of Chapter 4
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 86/87
86
Today¶s class was brought to you
by
Colgate
the toothpaste recommended by9 out of every 10 Dentists.
8/8/2019 MAT2020 - Unit 1 - Measures of Central Tendency and Dispersion
http://slidepdf.com/reader/full/mat2020-unit-1-measures-of-central-tendency-and-dispersion 87/87
The End
Recommended