Summary Managerial Statistics
Lisa de Vries
University of Groningen
Pre-MSc Marketing 2020-2021
,Table of contents
Chapter 1. What is statistics?.................................................................................................................3
Chapter 2. Graphical descriptive techniques I........................................................................................4
Chapter 3. Graphical descriptive techniques II.......................................................................................6
Chapter 4. Numerical descriptive techniques.......................................................................................10
Chapter 6. Probability...........................................................................................................................17
Chapter 7. Random variables and discrete probability distributions....................................................22
Chapter 8. Continuous probability distributions...................................................................................26
Chapter 9. Sampling distributions........................................................................................................30
Chapter 10. Introduction to estimation................................................................................................33
Chapter 11. Introduction to hypothesis testing....................................................................................36
Chapter 12. Inference about a population...........................................................................................38
Chapter 13. Inference about comparing two populations....................................................................43
..............................................................................................................................................................48
Chapter 15. Chi-Squared Tests.............................................................................................................50
Chapter 16. Simple linear regression and correlation..........................................................................53
Chapter 17. Multiple Regression..........................................................................................................58
Chapter 19. Nonparametric statistic.....................................................................................................63
When to use which test?......................................................................................................................67
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,Chapter 1. What is statistics?
Introduction
Statistics is a way to get information from data.
Descriptive statistics deals with methods of organizing, summarizing, and presenting data in a
convenient and informative way for a group that you choose. Descriptive statistics can use two
techniques:
1) Graphical techniques: allows statistics practitioners to present data in ways that make it easy for
the reader to extract useful information.
2) Numerical techniques: computing with real numbers; i.e. calculating the average or mean.
The technique we use depends on what specific information we would like to extract.
Inferential statistics is a body of methods used to draw conclusions or inferences about
characteristics of populations based on sample data. It’s the translation of limited information from
the sample to the entire population. So usually we don’t observe the entire population, because it’s
way too costly and often impossible. We only observe a few observations of the entire population.
Sample should be random or representative (everyone equally likely to be included). It involves
estimation and hypothesis testing.
A sample that is only a small fraction of the size of the population can lead to correct inferences only
a certain percentage of the time. You will find that statistics practitioners can control that fraction
and usually set it between 90% and 99%.
1.1 Key statistical concepts
Population = the group of all items of interest to a statistics practitioner. It does not necessarily refer
to a group of people. A descriptive measure of a population is called a parameter. It can be the
average, can be the spread, can be anything that describes the population. In most applications of
inferential statistics the parameter represents the information we need.
Sample = a set of data drawn from the studied population. It might be too tedious (and sometimes
impossible) to measure the parameter of the whole population, so we select a “sample”. A
descriptive measure of a sample is called a statistic. We use statistics to make inferences about
parameters.
Statistical inference = the process of making an estimate, prediction, or decision about a population
based on sample data. Because populations are almost always very large, investigating each member
of the population would be impractical and expensive. It’s far easier and cheaper to take a sample
from the population of interest and draw conclusions or make estimates about the population on the
basis of information provided by the sample.
However, such conclusions and estimates are not always going to be correct, because the
information is limited and we can’t translate it back to the entire population. For this reason we build
into the statistical inference a measure of reliability. There are two such measures:
1) The confidence level; the proportion of times that an estimating procedure will be correct
2) The significance level; when the purpose of the statistical inference is to draw a conclusion about
a population, the significance level measures how frequently the conclusion will be wrong.
Together these levels will be 100%, so if you know that you have a confidence level of 95%, you know
that the significance level is 5%.
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,Chapter 2. Graphical descriptive techniques I
Introduction
Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that
useful information is produced. Its methods make use of graphical techniques and numerical
descriptive measures to summarize and present the data, allowing managers to make decisions
based on the information generated.
The two most important factors that determine the appropriate descriptive technique to use are (1)
the type of data and (2) the information that is needed.
2.1 Types of data and information
A variable is some characteristic of a population or sample. It’s a number or a label used to describe
observations in a population or sample. We usually represent the name of the variable using
uppercase letters such as X, Y and Z. The values of the variable are the possible
observations/outcomes of the variable.
Data are the observed values of a variable. We will extract the information we seek from the right
data. There are three types of data:
1) Nominal data (= qualitative or categorical data): the values are categories. The values aren’t
numbers, but words that describe the categories. All we can say is that the categories are different,
but they have no order. If we observe for example the colour of the shirt that you’re wearing, the
colours are different, but there’s not a specific order in that green should be better/nicer than
yellow. No other calculation than determining frequencies are permitted here.
2) Ordinal data: appear to be nominal, but the difference is that the order of their values has
meaning. We can order the traits, but we can’t say how “much more” one category is than another.
By high education and low education there’s a difference in order, but we can’t exactly say how much
“more” a higher education is, it’s hard to express in numbers for example. Thus, it’s not the
magnitude of the values that’s important, it’s their order. Permissible calculations here are ones that
rank the data; frequencies and quantiles.
3) Interval data (= quantitative or numerical data): are real numbers, such as heights, weights,
incomes and distances. Values of interval data are consistent and meaningful. The differences
between outcomes matter. All calculations are allowed here; arithmetic, quantiles, frequencies.
Frequencies = the number of time a value occurs. Quantiles = defines a particular part of a data set,
i.e. a quantile determines how many values in a distribution are above or below a certain limit.
Arithmetic = The basic calculations we make in everyday life: addition, subtraction, multiplication and
division.
! We can treat higher-level data types as lower-level data types, but vice versa CAN’T.
2.2 Describing a set of nominal data
We can summarize (nominal) data in a table, which represents the categories and their counts
(= frequency distribution).
“A distribution in statistics is a function that shows the possible values for a variable and how often
they occur.” And then we can construct a relative frequency distribution by converting the
frequencies into proportions with which each occurs (frequency as part of the total).
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,Two graphical techniques can be used to display the results shown in a table:
1) A bar chart is often used to display frequencies and is created by drawing a rectangle
representing each category. The height of the rectangle represents the frequency. The
base is arbitrary.
2) A pie chart graphically shows relative frequencies. If we wish to emphasize the relative
frequencies instead of drawing the bar chart, we draw a pie chart. A pie chart is simply a
circle subdivided into slices that represent the categories. It is drawn so that the size of
each slice is proportional to the percentage corresponding to that category.
There are no specific graphical techniques for ordinal data. Consequently, when we wish to describe
a set of ordinal data, we will treat the data as if they were nominal and use the techniques described
in this section. The only criterion is that the bars in bar charts should be arranged in ascending (or
descending) ordinal values; in pie charts, the wedges are typically arranged clockwise in ascending or
descending order.
Two factors that identify when to use frequency and relative frequency tables, bar and pie charts:
1) The objective: describe a single set of data
2) The data type: nominal or ordinal
2.3 Describing the relationship between two nominal variables and
comparing two or more nominal data sets
Univariate = techniques applied to single sets of data. A type of data which consists of observations
on only a single characteristic or attribute.
Bivariate = techniques applied to depict the relationship between variables.
A cross-classification table (also: cross-tabulation table) is used to
describe the relationship between two nominal variables. The same
technique is used to compare two or more sets of nominal data.
To describe the relationship between two nominal variables, we
must remember that we are permitted only to determine the
frequency of the values. As a first step, we need to produce a cross-classification table that lists the
frequency of each combination of the values of the two variables.
An easy way to see if the two variables are related is to convert the frequencies in each row (or
column) to relative frequencies in each row (or column). Another way is to create bar charts for the
data. If the two variables are unrelated, then the patterns exhibited in the bar charts or the cross-
classification table should be approximately the same. If some relationship exists, then some bar
charts and numbers in the classification table will differ from others.
Factors that identify when to use a cross-classification table:
1) The objective: describe relationship between two variables and compare two or more sets of data.
2) The data type: nominal
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, Chapter 3. Graphical descriptive techniques II
Introduction
In chapter 2 we introduced graphical techniques to summarize and present nominal data. In this
chapter we do the same for interval data.
3.1 Graphical techniques to describe a set of interval data
The most important of these graphical methods is the histogram. An histogram is not only used to
summarize interval data, but also to help explain an important aspect of probability.
Here we create a frequency distribution for interval data by counting the number of observations
that fall into each of a series of intervals, called classes, that cover the complete range of
observations. Mostly the intervals (classes) are equally wide, but this is not essential. And to create
the frequency distribution manually, we count the number of observations that fall into each
interval.
Although the frequency distribution provides information about how the numbers
are distributed, the information is more easily understood and imparted by drawing
a picture or graph. The graph is called a histogram. A histogram is created by
drawing rectangles whose bases are the intervals and whose heights are the
frequencies.
The number of class intervals we select depends entirely on the number of observations in the data
set. The more observations we have, the larger the number of class intervals we need to use to draw
a useful histogram.
There’s a table which provides guidelines on choosing the number of classes:
Number of observations Number of classes
Less than 50 5-7
50-200 7-9
200-500 9-10
500-1,000 10-11
1,000-5,000 11-13
5,000-50,000 13-17
More than 50,000 17-20
So if we have 200 observations, the table tells us to use 7, 8, 9 or 10 classes.
An alternative is to use Sturges’s formula:
Number of class intervals = 1 + 3.3 log (n)
The approximate width of the classes can be determined by:
Largest observation−smallest observation
Class width =
Number of classes
The only condition we apply is that the first class interval must contain the smallest observation.
For example: class 1: amounts that are less than or equal to 15 (0 to 15), class 2 is then: amounts that
are MORE than 15 but less than or equal to 30. Class 3: MORE than 30 but less than or equal to 45
etc. NOTE: the table and formula are guidelines only. It is more important to choose classes that
are easy to interpret. So classes with intervals of 5 instead of 6 for example.
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