1. OAD30763 Statistics in Business and Economics
Week 2
Dr. Jenne Meyer
OAD30763 Statistics in Business and Economics

2. Visual Representations of Data

3. Histogram

4. Dot Plot
In a Dot Plot, each observation is plotted as a point on a single, horizontal axis. The axis is scaled so that each of the data points can be located uniquely on the axis. When there is more than one observation with the same value the points are “stacked” on top of each other.

5. Pareto Diagram
A Pareto Diagram is a bar chart in which the categories are plotted in order of decreasing relative frequency. In addition to the bars, the cumulative relative frequency of the categories is plotted on the same graph.

6. Pie Chart
A Pie Chart represents data in the form of slices or sections of a circle. Each slice represents a category and the size of the slice is proportional to the relative frequency of the category.

7. Frequency Distribution
A tabulation of n data values into k classes called bins, based on values of the data. The bin limits are cutoff points that define each bin. Bins must have equal widths and their limits cannot overlap.

8. Frequency curves
Represents the proportion/percentage of the population that fall into a certain range.
high slope = high frequency
low slope = low frequency

9. Skew Mode < Median < Mean Mean < Median < Mode
Positively skewed Negatively skewed
Skewed to the right Skewed to the left

10. Line or Bar Charts

11. Scatterplot

12. Pictograms

13. Frequency Distribution
A frequency distribution is a tabular summary of data showing the frequency (or number) of items in each of several non-overlapping classes.
The objective is to provide insights about the data that cannot be quickly obtained by looking only at the original data.

14. Frequency Distribution
Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 guests
Above Average
Below Average
Poor
Average
Below Average
Above Average
Average
Average
Above Average
Below Average
Poor
Excellent

15. Frequency Distribution
Poor
Below Average
Average
Above Average
Excellent
Rating 2 3 5 9 1
Total

16. Relative Frequency and Percent Frequency Distributions
Poor
Below Average
Average
Above Average
Excellent
Rating
.10
.15
.25
.45
.05
Total 10 15 25 45 5
100
.10(100) = 10
1/20 = .05

17. Crosstabulations and Scatter Diagrams
Thus far we have focused on methods that are used to summarize the data for one variable at a time.
Often a manager is interested in tabular and graphical methods that will help understand the relationship between two variables.
Crosstabulation and a scatter diagram are two methods for summarizing the data for two variables simultaneously.

18. Crosstabulation
The number of Finger Lakes homes sold for each style and price for the past two years is shown below.
quantitative
variable
categorical
variable
Home Style
Price
Range
Colonial Log Split A-Frame
Total 55 45
< $200,000
> $200,000
Total
100

19. Tabular and Graphical Methods
Data
Categorical Data
Quantitative Data
Tabular
Methods
Graphical
Methods
Tabular
Methods
Graphical
Methods
Bar Chart
Pie Chart
Frequency
Distribution
Rel. Freq. Dist.
Percent Freq.
Crosstabulation
Frequency
Distribution
Rel. Freq. Dist.
% Freq. Dist.
Cum. Freq. Dist.
Cum. Rel. Freq.
Cum. % Freq.
Crosstabulation
Dot Plot
Histogram
Ogive
Stem-and-
Leaf Display
Scatter
Diagram

20. Descriptive Statistics

21. Terminology Parameter
A number that describes the characteristic of the population
Statistic
A number that describes the behavior of the sample
Variable
A measured characteristic or attribute that differs for different subjects or people

22. Terminology Symbols  (Uppercase Sigma) = Summation
 (Mu) = Population mean
 (Lowercase Sigma) = Standard deviation
 (Pi) = Probability of success in a binomial trial
 (Epsilon) = Maximum allowable error
2 (Chi Square) = Nonparametric hypothesis test
! = Factorial
H0 = Null hypothesis
H1 = Alternate hypothesis

23. Measure of Central Tendency
A single value that summarizes a set of data. It locates the center of the values
Arithmetic mean
Weighted mean
Median
Mode
Geometric mean

24. ARITHMETIC MEAN ARITHMETIC MEAN
Pop mean = sum of all the values in pop
# of values in the pop
µ = ∑X N

25. Properties of arithmetic mean
Every set of interval data has a mean
All values are included
Mean is unique - only one
Useful to compare two or more populations
Sum of the deviations of each value from the mean will always be zero
Disadvantage of arithmetic mean
Mean may not be representative
Can’t use for open-ended (range) data

26. Median
The midpoint of the values (exactly half are below, half are above)
Used when the mean is not representative due to high value outliers
Unique number
Not affected by extremely large or small values
Can be used with open-ended range values
Can be used for several measurement types

27. Mode The value that appears most frequently
Can be used fir any measurement type
Not affected by extremely large or small values
Sometimes it doesn’t exist
Sometimes it represents more than one value

28. Formulas in Excel

29. Skewness – Mean, Median, Mode

30. Median of grouped data Median = L + n/2 - CF (i) f
selling prices of Whitner Pontiac
Price # sold CF
12 –
15 –
18 –
21 –
24 –
27 –
30 –
Median = 18, / (3000) 17
= 18,
= 19,588

31. Measures of Dispersion
Range
Mean deviation
Variance
Standard deviation
Range = highest value – lowest value
Mean deviation – the arithmetic mean of the absolute values of the deviations from the mean
The # deviates of average x amount from the mean
Variance – the arithmetic mean of the squared deviations from the mean
Compare the dispersion of two or more sets of data
Standard deviation – the square root of the variance
represents the spread or variability of the data, the average range from the center point

32. Variation
Population variation
=varp(…)
Sample variation
=var(…)

33. Standard Deviation Population variation Sample variation =stdevp(…)

34. Sample Standard Deviation
Sample standard deviation is most common use of statistics

35. Standard Deviation Example: Numbers Mean Standard Deviation
100,100,100,100,100,
90, 90, 100, 110,
Computing the standard deviation:
find the mean (100)
find the deviation/variance of each value form the mean (-10, -10, 0, 10, 10)
square the deviations/variances (100, 100, 0, 100, 100)
sum the squared deviations ( = 400)
divide the sum by the # of values minus 1 (# of values = 5 – 1 = 4, /4 = 100)
take the square root of the variance (10)
(Will be important in research when you are trying to determine the range of information.)

36. Coefficient of Variation
To compare dispersion in data sets with dissimilar units of measurement (e.g., kilograms and ounces) or dissimilar means (e.g., home prices in two different cities) we define the coefficient of variation (CV), which is a unit-free measure of dispersion:

37. Frequency curves
Normal distribution

38. Sample Variance, Standard Deviation, And Coefficient of Variation
the standard
deviation is
about 11%
of the mean
Coefficient of Variation

39. Formulas in Excel

40. Central Limit Theorem Chebyshev’s Theorem
If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. (the larger the sample, the more it appears to be a normal standard distribution)

41. Central Limit Theorem Chebyshev’s Theorem

42. Central Limit Theorem Chebyshev’s Theorem

43. Standard Normal Distribution
Z value – converts the actual distribution to a standard distribution. (It is the distance between the selected value (x) and the mean (µ) divided by the standard deviation (σ). It denotes the number of standard deviations a data value x is from the mean.
Normal distributions can be transformed to standard normal distributions by the formula:
A “Z” score always reflects the number of standard deviations above or below the mean a particular score is
A person scored 60 on a test with a μ=50 and σ=10, then he scored 1 standard deviations above the mean. Converting the test score to a Z score, an X of 70 would be:
Z=1=0.3413

44. Standard Normal Distribution
Standard Normal Table (once z is computed)
A table of probabilities for a Z random variable.
See page 479
5/18/2019

45. Example p 224/5, likelihood of finding a foreman w/ a salary between $1000 and $1100 is 34.13%

47. Standard Normal Distribution
p227
5/18/2019

48. Normal Distribution Examples
Chapter 3,
p 107 problem 27
Problem 29, 30, 31

49. Discussion
Key learnings?
Next weeks assignments.