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

Visual Representations of Data

Histogram

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.

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.

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.

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.

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

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

Line or Bar Charts

Scatterplot

Pictograms

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.

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

Frequency Distribution Poor Below Average Average Above Average Excellent Rating 2 3 5 9 1 Total 20

Relative Frequency and Percent Frequency Distributions Poor Below Average Average Above Average Excellent Rating .10 .15 .25 .45 .05 Total 1.00 10 15 25 45 5 100 .10(100) = 10 1/20 = .05

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.

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 18 6 19 12 55 45 < $200,000 > $200,000 12 14 16 3 Total 30 20 35 15 100

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

Descriptive Statistics

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

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

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

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

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

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

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

Formulas in Excel

Skewness – Mean, Median, Mode

Median of grouped data Median = L + n/2 - CF (i) f selling prices of Whitner Pontiac Price # sold CF 12 – 15 8 8 15 – 18 23 31 18 – 21 17 48 21 – 24 18 66 24 – 27 8 74 27 – 30 4 78 30 – 33 2 80 Median = 18,000 + 80/2 - 31 (3000) 17 = 18,000 + 1588 = 19,588

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

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

Standard Deviation Population variation Sample variation =stdevp(…)

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

Standard Deviation Example: Numbers Mean Standard Deviation 100,100,100,100,100,100 100 0 90, 90, 100, 110, 110 100 10 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 (100+100+0+100+100 = 400) divide the sum by the # of values minus 1 (# of values = 5 – 1 = 4, 400/4 = 100) take the square root of the variance (10) (Will be important in research when you are trying to determine the range of information.)

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:

Frequency curves Normal distribution

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

Formulas in Excel

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)

Central Limit Theorem Chebyshev’s Theorem

Central Limit Theorem Chebyshev’s Theorem

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

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

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

Standard Normal Distribution p227 5/18/2019

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

Discussion Key learnings? Next weeks assignments.