Chapter Six

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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.
BurnsMorningside College
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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.

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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 s2.
• 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).

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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.

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Measures of Variability

• Variability indicates the spread in scores.
• To calculate the variance
• Square the deviate scores S(X M)2
• And then divide by the number of scores (N)

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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.

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Raw Score Formula to Calculate Variance
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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 vvariance
• 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.

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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.

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Calculating and Interpreting the Standard
Deviation
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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.

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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.

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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

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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

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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.

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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

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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.

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Confidence Intervals

• Calculation of the two confidence intervals
  consists of
• 95 Confidence Interval
• 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
• 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.