Measures of Dispersion

1
Measures of DispersionThe Standard Normal
Distribution

• 2/5/07

2
The Semi-Interquartile Range (SIR)

• A measure of dispersion obtained by finding the
  difference between the 75th and 25th percentiles
  and dividing by 2.
• Shortcomings
• Does not allow for precise interpretation of a
  score within a distribution
• Not used for inferential statistics.

3
Calculate the SIR

• 6, 7, 8, 9, 9, 9, 10, 11, 12
• Remember the steps for finding quartiles
• First, order the scores from least to greatest.
• Second, Add 1 to the sample size.
• Third, Multiply sample size by percentile to find
  location.
• Q1 (10 1) .25
• Q2 (10 1) .50
• Q3 (10 1) .75
• If the value obtained is a fraction take the
  average of the two adjacent X values.

4
Variance (second moment about the mean)

• The Variance, s2, represents the amount of
  variability of the data relative to their mean
• As shown below, the variance is the average of
  the squared deviations of the observations about
  their mean

• The Variance, s2, is the sample variance, and is
  used to estimate the actual population variance,
  s 2

5
Standard Deviation

• Considered the most useful index of variability.
• Can be interpreted in terms of the original
  metric
• It is a single number that represents the spread
  of a distribution.
• If a distribution is normal, then the mean plus
  or minus 3 SD will encompass about 99 of all
  scores in the distribution.

6
Definitional vs. Computational

• Definitional
• An equation that defines a measure
• Computational
• An equation that simplifies the calculation of
  the measure

7
Calculating the Standard Deviation
8
Interpreting the standard deviation

• We can compare the standard deviations of
  different samples to determine which has the
  greatest dispersion.
• Example
• A spelling test given to third-grader children
• 10, 12, 12, 12, 13, 13, 14
• xbar 12.28 s 1.25
• The same test given to second- through
  fourth-grade children.
• 2, 8, 9, 11, 15, 17, 20
• xbar 11.71 s 6.10

9

• Interpreting the standard deviation
• Remember
• Fifty Percent of All Scores in a Normal Curve
  Fall on Each Side of the Mean

10
Probabilities Under the Normal Curve
11
The shape of distributions

• Skew
• A statistic that describes the degree of skew for
  a distribution.
• 0 no skew
• or - .50 is sufficiently symmetrical
• value skew
• - value - skew
• You are not expected to calculate by hand.
• Be able to interpret

12
Kurtosis

• Mesokurtic (normal)
• Around 3.00
• Platykurtic (flat)
• Less than 3.00
• Leptokurtic (peaked)
• Greater than 3.00
• You are not expected to calculate by hand.
• Be able to interpret

13
The Standard Normal Distribution

• Z-scores
• A descriptive statistic that represents the
  distance between an observed score and the mean
  relative to the standard deviation

14
Standard Normal Distribution

• Z-scores
• Convert a distribution to
• Have a mean 0
• Have standard deviation 1
• However, if the parent distribution is not normal
  the calculated z-scores will not be normally
  distributed.

15
Why do we calculate z-scores?

• To compare two different measures
• e.g., Math score to reading score, weight to
  height.
• Area under the curve
• Can be used to calculate what proportion of
  scores are between different scores or to
  calculate what proportion of scores are greater
  than or less than a particular score.
• Used to set cut score for screening instruments.

16
Class practice

• 6, 7, 8, 9, 9, 9, 10, 11, 12
• Calculate z-scores for 8, 10, 11.
• What percentage of scores are greater than 10?
• What percentage are less than 8?
• What percentage are between 8 and 10?

17
Z-scores to raw scores

• If we want to know what the raw score of a score
  at a specific tile is we calculate the raw using
  this formula.
• With previous scores what is the raw score
• 90tile
• 60tile
• 15tile

18
Transformation scores

• We can transform scores to have a mean and
  standard deviation of our choice.
• Why might we want to do this?

19
With our scores

• We want
• Mean 100
• s 15
• Transform
• 8 10.

20
Key points about Standard Scores

• Standard scores use a common scale to indicate
  how an individual compares to other individuals
  in a group.
• The simplest form of a standard score is a Z
  score.
• A Z score expresses how far a raw score is from
  the mean in standard deviation units.
• Standard scores provide a better basis for
  comparing performance on different measures than
  do raw scores.
• A Probability is a percent stated in decimal form
  and refers to the likelihood of an event
  occurring.
• T scores are z scores expressed in a different
  form (z score x 10 50).

21
Examples of Standard Scores