PSY 307 – Statistics for the
Chapter 3-5 – Mean, Variance,
Standard Deviation and Z-scores
Measures of Central Tendency
Mode – the most frequently occurring
Median – the middle value in the data
Mean – average
Mode – can always be used
Median – can sometimes be used
Mean – can never be used
The value of the most frequently
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
The middle value when observations
are ordered from least to most, or
Half the numbers are higher and half
When there is an even number of
observations, the median is the
average of the two middle values.
The most commonly used and most
Mean = sum of all observations
number of all observations
Observations can be added in any
Sample vs population
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
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
The mean is the preferred average.
Mean and modal ranks are not
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).
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
Never report a median with
Never report the mean.
Measures of Variability
Range – difference between highest
and lowest value.
Variance – the mean of the squared
deviations (differences) from the
Standard Deviation – square root of
The average amount that observations
deviate from the mean.
Interquartile Range (IQR)
The range for the middle 50% of
Distance between the 25th and 75th
Remove the highest and lowest
25% of scores then calculate the
range for the remaining values.
Used because it is insensitive to
Using IQR (from Holcomb)
In Rio, what percentage had been
injecting from 4.5 to 14 years?
Median Year Injecting = 10
IQR is 4.5-14 (from text).
0 4.5 14
25% 25% 25% 25% 100%
Median = 50%
Sample variance = S2
Population variance = s2
Sample standard deviation = S or
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
Source of example using dogs: http://www.mathsisfun.com/standard-deviation.html
Next, compare their heights to the
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
σ2 = 2062 + 762 + (-224)2 + 362 + (-94)2 = 108,520 = 21,704
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.
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
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
This generalization is true no matter
what the shape of the distribution.
It works for skewed distributions.
A Measure of Distance
The mean shows the position of the
balance point within a distribution.
The standard deviation is a unit of
distance that is useful for
Standard deviations cannot have a
They can measure in both positive and
negative directions from the mean.
Definition formula – easier to
s ( X X )2 ( X X )2
The numerator is also called the
Sum of the Squares (squared
differences), abbreviated SS
Computation formula – easier to
use, especially with large data sets.
s2 X 2 ( X ) 2
The computational and definition
formulas produce the same result.
Population vs Sample
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
Using n-1 more accurately
estimates the variability in a
Variance for sample:
Variance for population:
Indicates how many SDs an
observation is above or below the
mean of the normal distribution.
Formula for converting any score to
Z= X – m 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
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%).
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
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
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)