Statistics for Decision Making

Descriptive Statistics

QM 2113 -- Fall 2003

Instructor: John Seydel, Ph.D.

Student Objectives

Locate course materialsSummarize course conceptsDistinguish between the two primary types of dataSummarize typical descriptive measures used for the different data typesPerform basic descriptive statistics calculationsCreate visual summaries of univariate data

Let’s Start With a Case

Read the Web Analytics caseLet’s look at the revenue figuresHow much revenue does a website earn?Hard question to answer: there’s variability!But not all is lost; what’s a reasonable to answer the first question?Yes, the average!But . . .

Dealing With the VariationWe can’t (or at least shouldn’t) ignore that there are differences among those valuesLook at a different set of data: Same average (almost) But there’s a lot of difference!

So, what can we say about the variation among the values?How about summarizing it with the range?A better figure: the standard deviation Essentially, it’s the average difference among

the values Not so easy to calculate That’s why we have computers!

Boiling Down Many Numbers to a Few

Thus we can typically use two values to summarize a set of data The average (i.e., the mean) The standard deviation

Easy to calculate with Excel =AVERAGE() =STDEV()

Can we do this with all data?Consider another collection of observations, different from but related to the first . . .

Let’s Look at the Business Models

How are these data different than the revenue data?Yes, these aren’t numbersInstead, they’re categoriesOK, so how would we describe the business model used by these firms?Again, yes, percentagesNow, can we summarize the variation?No . . . !

We Now Need to Summarize What We’ve Experienced

There will always be variation, but we can still concisely describe large amounts of information

What the values tend to be The variation among the values

Two types of data Numeric (e.g., revenues) Categorical (e.g., business models)

We can summarize numeric data according to General tendency (e.g., the average) How different they are (e.g., the standard deviation)

But the best we can do with categorical data is to list the percentages by categoryThe difference: information content

What is statistics? Description (Data analysis) ---> Stage 1 Inference (Applying results) ---> Stage 2

Data types Numeric (aka “quantitative”) Categorical (aka “qualitative”)

Descriptive analysis Informal (tables & charts) Summary measures

Inference: we’ll get there in a few weeks

Now, Let’s Do an Overview

Type of analysis depends upon data: Numeric; you’ll also see these terms

Ratio Interval Ordinal

Categorical; you’ll also see these terms Ordinal Nominal

Other examples?

Very Important

Three general forms Informal

Tables Charts

Formal: numeric (i.e., statistics)

Forms basis for performing inferential analyses

Descriptive Analysis

Categorical data Percentages Analysis of proportions

Numeric data Single numbers that summarize

Location (i.e., general tendencies) Variation (i.e., how different the values are)

Primary importance Mean Standard deviation

Descriptive Statistics (Formal)

Mean -- just a simple average Add the values and divide by number of

observations

Standard deviation Average difference among the values Process:

Subtract the average from each value Square each result “Average” the squared results Take the square root of that result

Primary Measures

Less important but need to be familiar with: Location

Median Mode Quantiles

Variation Range Min and Max

Both (?) Z-score Empirical Rule

We’ll revisit these in a week or so

Miscellaneous Statistics

Getting organized: Ordered array Frequency distribution

Absolute frequencies Relative frequencies (%) Cumulative frequencies

Cumulative relative frequencies

Histogram (frequencies)Other Stem-leaf display Ogive (cumulative frequencies)

Numeric Data: Charts & Tables

Start by breaking the data range into k equal width intervals Let n represent the number of observations Number of intervals such that 2k > n

Interval width Start with: (Max - Min) / k Use convenient breakpoints for intervals

91.0 through 97.4 (OK) 90.0 through 95.0 (Better)

Intervals: no overlap; no gaps

Frequency Distributions: Determining Frequency Groups

“Absolute” frequenciesCount number of observations in each interval

Relative frequenciesDivide absolute frequency by total number of observations

Cumulative frequenciesAdd frequencies for all previous intervals (note difference from manner done in text)

Cumulative relative frequenciesAdd relative frequencies for all previous intervals

Frequency Distributions: Determining Frequencies

What are they? Just graphical displays of frequency

distributions: absolute, relative, cumulative Provide “picture” of the variation in the

dataWhat they are not: Bar charts But they do for numeric data what bar

charts do for categorical dataBasics Horizontal axis: values for variable of

concern Vertical axis: indicates corresponding

frequencies

Histograms

What is statistics all about?It’s about dealing with variation Summarizing information (description) Making decisions based upon that

summarizationType of analysis depends on data type Numeric Categorical

Description Formal

Numeric data: average and standard deviaiton Categorical data: percentages

Informal: frequency tables and charts data

Summary

Appendix

Sampling

Population

Sample

Parameter

Statistic

Schematic View

Probability is w hat a llow s the linkage betw een descriptive and inferentialanalyses

Inform al

Inferential Analyses

Sum m ary M easures

Num eric Data

Inform al

Inferential Analyses

Sum m ary M easures

Categorical Data

Statistics