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Statistics and Research methods
Wiskunde voor HMI
Betsy van Dijk
Introduction

Statistics is about
–
–
–
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Systematically studying phenomena in which we are
interested
Quantifying variables in order to use mathematical
techniques
Summarizing these quantities in order to describe
and make inferences
Using these descriptions and inferences to make
decisions or understand
The Two Branches of
Statistical Methods

Descriptive statistics (beschrijvende statistiek)
–

Used to summarize, organize and simplify data
Inferential statistics (toetsende statistiek)
–
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Draw conclusions/make inferences that go beyond
the numbers from a research study
Techniques that allow us to study samples and then
make generalizations about the populations from
which they were selected
Descriptive Statistics

Numbers that describe the characteristics of a
particular data set
–
–
“The average age in the class is 27 years”
“The range of ages in class is 22 years, from a
minimum of 20 to a maximum of 42”
Inferential Statistics

Descriptive statistics from a sample that are
used to make inferences about the
characteristics of a population.
–
“The average age of people taking Research
Statistics is 27 years.”
a “parameter”
People taking
Research Statistics
A sample of people taking
Research Statistics
Basic Concepts - Variables

Things that change
–
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
Environmental events or conditions
Personal characteristics or attributes
Behaviors
Anything that takes on different values in
different situations (even just through time)
Basic Concepts

Value
–

Score
–

A particular person’s value on a variable
Data
–

A possible number or category that a score can have
Scores or measurements of phenomena, behaviors,
characteristics, etc.
A Statistic
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A number that summarizes a set of data in some way
Populations and Samples

Population
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
Sample
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
Set of all the individuals of interest in a population study
Set of individuals selected from the population
Sampling error
–
Discrepancy, or amount of error that exists between the
sample statistic and population parameter
Measurement


Measurement is the process of assigning
numbers to variables following a set of rules
There are different levels of measurement
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Nominal
Ordinal
Interval
Ratio
Nominal Measurement




Places data in categories
Non-quantitative (e.g. qualitative), even though there
might be numbers involved
Nominal (categorical) variables
Examples
–
Male/Female

–
M,F (0,1)
Voting precinct

Alucha, Dade, Palm Beach (023, 095, 167)
Ordinal Measurement





Places data in order
Quantitative as far as ranking goes
Rank-order (ordinal) variables
Distance between values varies
Examples
–
First, second, third

–
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(1,2,3) (2.7, 2.8, 7.6)
Young, Middle Age, Old
Very Good, Good, Intermediate, Bad, Very Bad
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(1,2,3,4,5)
Interval Measurement


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Has all the characteristics of ordinal data
Additionally, the differences between values represents
a specific amount of whatever is being measured
(equal intervals represent equal amounts)
Examples
–

Temperature (the difference between 20C and 40C is the
same as 60C and 80C, but 0 is not the absence of
temperature)
Note: Many rating scales are treated like interval
measurements
Ratio Measurement


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Has all the characteristics of interval data
Additionally, has a true zero which represents the
absence of whatever is being measured
Examples
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
Time (e.g. reaction time)
Distance
The zero point allows you to make statements about
ratios (e.g. 100 feet is twice as far as 50 feet)
A Few More Things

Continuous variables
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Take on an infinite number of values between two
measured levels (e.g. time measurements)
Discrete variables
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Have no intermediate values (e.g. number of
people in class)
Math Warm-Up

Order of operations
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Proportion
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Some portion of some total amount
Expressed by a fraction or a decimal
To calculate, divide the portion by the total amount
Percentage
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Parentheses, exponents, multiplication/division, addition/subtraction
PEMDAS, or “please excuse my dear aunt sally”
Summation using the summation statistic before other addition/substraction
A proportion that is scaled to be out of 100 (instead of some other total amount)
To calculate, first calculate the proportion, then multiply by 100
Mathematical operators
–
Exponents, square roots, parentheses, summation, indexing
Math Warm-Up

Practice problems
y  ax  b
( x  b) 2
y
a
N
y   xi  b
i 1
y
2
(
x

b
)

a
Frequency Tables

Used to summarize data

Steps in making a frequency table
1. Make a list of each possible value
2. Count up the number of scores with each value
3. Make a table

Frequency table shows how often each value
occurs
A Frequency Table
Stress
Rating
Frequency
Percent
10
9
8
7
6
5
4
3
2
1
0
14
15
26
31
13
18
16
12
3
1
2
9.3
9.9
17.2
20.5
8.6
11.9
10.6
7.9
2.0
0.7
1.3
Histogram -- Stress-rating Data
0
1
2
3
4
5
6
7
8
9
10
Frequency
2
1
3
12
16
18
13
31
26
15
14
35
30
25
Frequency
Stress
20
15
10
5
0
0
1
2
3
4
5
6
Stress Rating
7
8
9
10
Grouped Frequency Table

A frequency table that uses intervals
Stress
Rating Interval
Frequency
Percent
10-11
8-9
6-7
4-5
2-3
0-1
14
41
44
34
15
3
9
27
29
23
10
2
Frequency Graphs
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Histogram
Frequency Graphs
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Frequency
polygon
Shapes of Frequency Distributions
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Unimodal, bimodal, and
rectangular
Shapes of Frequency Distributions

Unimodal – there is a single most frequent
value or “peak”

Bimodal – there are two most-frequent values
or peaks

Rectangular – there is no peak; all values are
about equally frequent
Shapes of Frequency Distributions

Symmetrical and skewed distributions
Shapes of Frequency Distributions
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
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Symmetrical – left and right halves of the distribution
have approximately the same shape
Skewed – left and right halves of the distribution do not
have the same shape
“skew” is towards the side with the fewer cases


Right (or positive) skew = few cases with large scores
Left (or negative) skew = few cases with small scores
Skewed distributions may be caused
by:

“Ceiling effects” – limitation in the high end of
the scale

“Floor effects” – limitation in the low end of the
scale
Sometimes skewed distributions occur
because of the nature of the variable
itself…
Millions of Families
35
30
25
20
15
10
5
0
0
1
Number of Children
2
Shapes of Frequency Distributions
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Normal and kurtotic distributions
Measures of Central Tendency
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Median
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Mode
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The value in the middle
The most common value
Mean
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The average value
The Mean
X

M
N
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

M = the mean
X = the scores
N = the number of scores
The Median
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Rank the scores from lowest to highest
Median is the score in the middle
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if even number of scores, by convention take the
average of the two middle ones
Median is not as sensitive to extreme values as
the mean
The Mode
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The most frequent score
To compute the mode: look at a frequency
table and find the most frequent score.
In a symmetrical, unimodal distribution, the
mean, median and mode are all the same.
Symmetrical Distribution
F
r
e
q
u
e
n
c
y
3,5
3
2,5
2
1,5
1
0,5
0
4
5
6
Mean
Median
Mode
7
8
Question
Negative Skew
F
r
e
q
u
e
n
c
y
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
4
5
6
7
Where (approximately) will Mean, Median and Mode be situated?
8
Problem with the Mean
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The mean can be strongly influenced by outliers
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This distorts the mean as a measure of central tendency
The median and mode are less affected by outliers
Measures of Variance
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A single number that tells you how spread out a
distribution is
8
8
7
7
6
Frequency
All M = 15.0
5
4
3
5
4
3
2
2
1
1
0
8
0
2.5
7.5
9
7
12.5 17.5 22.5 27.5
11
13
15
17
# of Chews
6
# of Chews
Frequency
Frequency
6
5
4
3
2
1
0
12
13
14
15
16
# of Chews
17
18
19
21
Measures of Variance

Range: difference between the maximum and
minimum observed values

Variance: a measure of the amount that values
differ from the mean of their distribution

Standard deviation: the average amount
(approximately) that values differ from the mean
of their distribution
Variance


Formula for the sample variance:
2
X  M


2
SD 
N
Estimate of the population variance:
SD


2
X  M



2
N 1
Unbiased estimate of population variance
Degrees of freedom: df = N-1
Describing Individual Values

Sometimes observations have values that people are
familiar with
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But sometimes values are on an unfamiliar scale
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Rating 1 to 10, Age, Temperature, SAT
Score on the Wisconsin Card Sorting Task
APGAR score
How can you communicate the relative value of a given
observation?
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Is that a very high value? very low? somewhere in the middle?
Z Scores

Characterize a score in relation to the distribution
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The number of standard deviations the score is
above or below the mean is called the Z score

Formula for Z score:
X M
Z
SD
Standard and Raw Scores

Z scores are also called “standard scores”

The original scores are called “raw scores”

For a distribution of Z scores, always M = 0

... and always SD = 1