Download Distribution of Means and the Standard Error of the Mean

Chapter Six
Summarizing and Comparing
Data: Measures of Variation,
Distribution of Means and the
Standard Error of the Mean, and z
Scores
PowerPoint Presentation created by
Dr. Susan R. Burns
Morningside College
Smith/Davis (c) 2005 Prentice Hall
Measures of Variability

Variability indicates the spread
in scores.
–
–
–
Range – the easiest measure
of variability to calculate; rank
order the scores in the
distribution and then subtract
the smallest score from the
largest to find the range
Range =
highest score – lowest score
The range does not provide
much information about the
distribution under
consideration.
Smith/Davis (c) 2005 Prentice Hall
Measures of Variability

Variability indicates the spread in scores.
–
Variance – is the single number that represents the total
amount of variability in the distribution.
 The variance of the sample is symbolized by S2,
whereas the variance of a population is symbolized by
σ2.
 The larger the number, the greater the total spread of
scores.
 The variance and standard deviation are based on how
much each score in the distribution deviates from the
mean, called the deviation score (X – M or x).
Smith/Davis (c) 2005 Prentice Hall
Measures of Variability

Variability indicates the spread in scores.
–
Variance – is the single number that represents the total
amount of variability in the distribution.
 If you sum the deviation scores, the total is zero
because the deviation scores are evenly distributed
above and below the mean.
 Because the mean is a balance point, the sum of the
deviation scores above the mean will always equal the
sum of the deviations below the mean, and thus will
cancel each other out.
Smith/Davis (c) 2005 Prentice Hall
Measures of Variability

Variability indicates the spread in scores.
– To calculate the variance:


Square the deviate scores Σ(X – M)2
And then divide by the number of scores (N)
Smith/Davis (c) 2005 Prentice Hall
Measures of Variability

Variability indicates the spread in scores.
–
Variance – is the single number that represents the total amount of
variability in the distribution.
 The authors mention that because this task is tedious and can
lead to calculation errors, the recommend using the raw score
formula with the following steps:
–
Square each raw score, then sum these squared scores
– Sum the raw scores; square this sum. Divide the squared sum by
N.
– Subtract the product obtained in Step 2 from the total obtained in
Step 1.
– Divide the product obtained in Step 3 by N.
Smith/Davis (c) 2005 Prentice Hall
Raw Score Formula to Calculate
Variance
Smith/Davis (c) 2005 Prentice Hall
Calculating and Interpreting the
Standard Deviation




To calculate the standard deviation (SD), all you do is
take the square root of the variance.
standard deviation = √variance
As with variance, the greater the variability or spread of
scores, the larger the stander deviation.
To understand the standard deviation, we must discuss
the normal distribution (also called the normal or the
bell curve).
In the normal curve, the majority of scores cluster
around the measure of central tendency with fewer and
fewer scores occurring as we move away from it.
Smith/Davis (c) 2005 Prentice Hall
The Normal Curve


The mean, median, and
mode all have the same
value and the
distribution is
symmetrical.
Distances from the
mean of a normal
distribution can be
measured in standard
deviation units.
Smith/Davis (c) 2005 Prentice Hall
Calculating and Interpreting the
Standard Deviation
Smith/Davis (c) 2005 Prentice Hall
Calculating and Interpreting the
Standard Deviation

When we describe the shape of a
normal distribution, how flat or
peaked it is describes its kurtosis.
There are three types of kurtosis:
– Leptokurtic distributions are
tall and peaked. Because the
scores are clustered around the
mean, the standard deviation will
be smaller.
– Mesokurtic distributions are
the ideal example of the normal
distribution, somewhere between
the leptokurtic and playtykurtic.
– Platykurtic distributions are
broad and flat.
Smith/Davis (c) 2005 Prentice Hall
Distribution of Means and the Standard
Error of the Mean

A distribution of means is where the “scores” in the
distribution are means, not scores from individual
participants with the following characteristics:
–
The distribution of means will approximate a normal
distribution if the population is a normal distribution or if
each sample you randomly select contains at least 30
scores. If the distribution of means is a normal distribution,
then we already have considerable information concerning
the percentage of scores falling the respective standard
deviations.
Smith/Davis (c) 2005 Prentice Hall
Distribution of Means and the Standard
Error of the Mean

A distribution of means is where the “scores” in the distribution
are means, not scores from individual participants with the
following characteristics:
–
The mean of a distribution of means will be the same as the mean
of the population. Using the Greek letter mu, μ, to symbolize the
mean of a population and μM to symbolize the mean of a
distribution of means, we can say that μM = μ.

You can calculate the variance for a distribution of means; you simply
treat each mean as a raw score and proceed as you would in
calculating any variance. Once you have calculated the variance of a
distribution of means, it is interesting to note that this variance is equal
to the population variance divided by the number in each of your
samples with the following formula:
Smith/Davis (c) 2005 Prentice Hall
Distribution of Means and the Standard
Error of the Mean

A distribution of means is where the “scores” in the
distribution are means, not scores from individual
participants with the following characteristics:
–
The mean of a distribution of means will be the same as the
mean of the population. Using the Greek letter mu, μ, to
symbolize the mean of a population and μM to symbolize the
mean of a distribution of means, we can say that μM = μ.

As with any other variance, the standard deviation of a
distribution of means is the square root of the variance of the
distribution of means with the following formulas:
Smith/Davis (c) 2005 Prentice Hall
Distribution of Means and the Standard
Error of the Mean

A distribution of means is where the “scores” in the
distribution are means, not scores from individual
participants with the following characteristics:
–
The standard deviation of a distribution of means is known
as the standard error of the mean (SEM), which tells you
how much variability you will find in the population from
which you drew the sample(s) in your research.
Smith/Davis (c) 2005 Prentice Hall
z Scores



z scores provide the
distance, in standard
deviation units, of a raw
score from the mean.
The formula for a z Scores is
as follows:
To convert a z score back to
a raw score you can use the
following formula:
X = (z)(SD) + M
Smith/Davis (c) 2005 Prentice Hall
Confidence Intervals




Statisticians have combined their knowledge of the
normal curve and an interest in the distributions of
sample means to produce another important concept:
confidence intervals
Interval estimates – are estimates of the range or interval
that is likely to include the population characteristic
(called a parameter), such as the mean or variance. An
interval that has a specific percentage associated with it
is called a confidence interval.
For the 95% confidence interval we are seeking limits that
will encompass a total of 95%, 47.5% above the mean,
47.5% below the mean.
The 99% confidence interval encompasses a total of
99%, 49.5% above and 49.5% below the mean.
Smith/Davis (c) 2005 Prentice Hall
Confidence Intervals

Calculation of the two confidence intervals consists
of:
–
95% Confidence Interval
1.
2.
3.
–
Multiply SD by –z score (-1.96) and +z score (+1.96)
Subtract the first product from the mean to establish the
lower confidence limit.
Add the second product to the mean to establish the upper
confidence limit.
99% Confidence Interval
1.
2.
3.
Multiply SD by –z score (-2.57) and +z score (+2.57)
Subtract the first product from the mean to establish the
lower confidence limit.
Add the second product to the mean to establish the upper
confidence limit.
Smith/Davis (c) 2005 Prentice Hall