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PSY 307 – Statistics for the
Behavioral Sciences
Chapter 3-5 – Mean, Variance,
Standard Deviation and Z-scores
Measures of Central Tendency
(Representative Values)

Quantitative data:




Mode – the most frequently occurring
observation
Median – the middle value in the data
Mean – average
Qualitative data:



Mode – can always be used
Median – can sometimes be used
Mean – can never be used
Mode

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

The value of the most frequently
occurring observation.
In a frequency distribution, look for
the highest frequency.
In a graph, look for the peaks or
highest bar in a histogram.
Distributions with two peaks are
bimodal (have two modes).

Even if the peaks are not exactly the
same height.
Median

The middle value when observations
are ordered from least to most, or
vice versa.


Half the numbers are higher and half
are lower.
When there is an even number of
observations, the median is the
average of the two middle values.
Mean


The most commonly used and most
useful average.
Mean = sum of all observations
number of all observations
=

X
n
Observations can be added in any
order.
Notation

Sample vs population






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Population notation = Greek letters
Individual value = x (lower case)
Sample mean = x or M
Population mean = m
Summation sign = 
Sample size = n
Population size = N
Mean as Balance Point

The sum of the deviations from the
mean always equals zero.


The mean is the single point of
equilibrium (balance) in a data set.
The mean is affected by all values
in the data set.


If you change a single value, the mean
changes.
Demo
The Most Descriptive Average


When a distribution is not skewed
(lopsided), the mean, median &
mode are similar.
When a distribution is skewed, the
mean is closer to the extreme
values, mode is farthest.


Report both the mean and median for a
skewed distribution.
The mean is the preferred average.
Ranked Data

Mean and modal ranks are not
informative.



The mean always equals the median
(middle) rank, so use the median.
The mode occurs when there is a tie in
the data, but doesn’t mean much.
Find the median by finding the
middle rank (or the average of the
two middle ranks).
Fedex Cup Rankings for Golfers
Player
Rank
Points
Walker
1
1650
Spieth
2
1409
Holmes
3
1233
Reed
4
1126
Watson
5
1088
Johnson
6
1005
Hoffman
7
948
Median = 1126
Fedex Cup Rankings for Golfers
Player
Rank
Points
Walker
1
1650
Spieth
2
1409
Holmes
3
1233
Reed
4
1126
Watson
5
1088
Johnson
6
1005
Hoffman
7
948
Streb
8
903
Median = (1126 + 1088)/2 = 1107
Qualitative Data Averages


The mode can always be used.
The median can only be used when
classes can be ordered.



The median is the category that
contains 50% in its cumulative
frequency.
Never report a median with
unordered classes.
Never report the mean.
Psychology Majors
Year
N
Cumulative Freq.
Freshmen
205
.30 or 30%
Sophomore
198
.59 or 59%
Junior
155
.82 or 82%
Senior
123
1.00 or 100%
Total
681
The median is the category that contains the middle
observation. The middle is at 50%.
The category containing that observation is Sophomore,
so Sophomore is the median.
Measures of Variability


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Range – difference between highest
and lowest value.
Variance – the mean of the squared
deviations (differences) from the
mean.
Standard Deviation – square root of
the variance.

The average amount that observations
deviate from the mean.
Interquartile Range (IQR)

The range for the middle 50% of
observations.

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
Distance between the 25th and 75th
percentiles.
Remove the highest and lowest
25% of scores then calculate the
range for the remaining values.
Used because it is insensitive to
extreme observations.
Using IQR (from Holcomb)

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In Rio, what percentage had been
injecting from 4.5 to 14 years?
Median Year Injecting = 10
IQR is 4.5-14 (from text).
IQR
0
4.5
25%
14
25%
25%
Median = 50%
25%
100%
More Notation
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Sample variance = S2
Population variance = s2
Sample standard deviation = S or
SD
Population standard deviation = s
Interquartile range = IQR
What Does Variance Describe?

Variance and standard deviation
describe the amount that actual
observations differ from the mean.



How spread out are the scores?
The range doesn’t tell us how
scores are distributed between the
high and low values.
Because the mean is the balance
point, the mean of the unsquared
deviations is always zero.
An example using dogs.
First calculate the height of the dogs.
Mean = 600 + 470 + 170 + 430 + 300 = 1970 = 394 mm
5
5
Source of example using dogs: http://www.mathsisfun.com/standard-deviation.html
Next, compare their heights to the
mean.
The green line shows the mean. Subtract the mean from
each dog’s height. Because some dogs are taller and
others are shorter, some of the differences will be positive
and some negative numbers. These differences will
cancel each other out because the mean is the balance
point in the distribution of dog heights.
Square the differences and take
the mean.
σ2 = 2062 + 762 + (-224)2 + 362 + (-94)2 = 108,520 = 21,704
5
5
Take the square root to return to
the original units of measure.


σ = √21,704 = 147
Which dogs are within one
standard deviation of the mean?
Rottweillers are unusally tall dogs. And Dachsunds
are a bit short.
Standard Deviation


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The variance is expressed in
squared units (e.g., squared lbs)
which are hard to interpret.
Taking the square root of the
variance expresses the average
deviation in the original units.
The square root of the variance
gives a slightly different result than
taking the average of the absolute
deviations.
Interpreting the SD

For most distributions, the majority
of observations fall within one
standard deviation of the mean.


A very small minority fall outside two
standard deviations.
This generalization is true no matter
what the shape of the distribution.

It works for skewed distributions.
A Measure of Distance

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The mean shows the position of the
balance point within a distribution.
The standard deviation is a unit of
distance that is useful for
comparing scores.
Standard deviations cannot have a
negative value.

They can measure in both positive and
negative directions from the mean.
Definition Formula

Definition formula – easier to
understand conceptually.
s

2
(
X

X
)

2
(
X

X
)

N
The numerator is also called the
Sum of the Squares (squared
differences), abbreviated SS
Computation Formula

Computation formula – easier to
use, especially with large data sets.
s2 

2
2
X

(
X
)


n
s
SS
N
The computational and definition
formulas produce the same result.
Population vs Sample


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The formulas are different
depending on whether a sample or
a population is being measured.
Use n-1 in the denominator when
using s or s2 to estimate s or s2 for
a population.
Using n-1 more accurately
estimates the variability in a
population.
Formulas

Variance for sample:
s2 

SS
n 1
Variance for population:
SS
s 
N
2
Z-Score


Indicates how many SDs an
observation is above or below the
mean of the normal distribution.
Formula for converting any score to
a z-score:
Z= X – m
s
m  mean
s  std. deviation
Properties of z-Scores

A z-score expresses a specific value
in terms of the standard deviation
of the distribution it is drawn from.


The z-score no longer has units of
measure (lbs, inches).
Z-scores can be negative or
positive, indicating whether the
score is above or below the mean.
Standard Normal Curve
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By definition has a mean of 0 and
an SD of 1.
Standard normal table gives
proportions for z-scores using the
standard normal curve.
Proportions on either side of the
mean equal .50 (50%) and both
sides add up to 1.00 (100%).
Other Distributions


Any distribution can be converted to
z-scores, giving it a mean of 0 and
a standard deviation of 1.
The distribution keeps its original
shape, even though the scores are
now z-scores.


A skewed distribution stays skewed.
The standard normal table cannot
be used to find its proportions.
Transformed Standard Scores

Z-scores are useful for converting
between different types of standard
scores:



IQ test scores, T scores, GRE scores
The z-scores are transformed into
the standard scores corresponding
to standard deviations (z).
New score = mean + (z)(std dev)