Describing Distributions, Measures of Central Tendency and Variability

1
Describing Distributions, Measures of Central
Tendency and Variability

• Hansel Burley

2
Describing distributions Shape, Central
Tendency, and Variation

• Shape You need information about its shape
• Normal, spread out, squeezed in, or skewed?
• Central tendency mode, median, mean
• Variation range, average(mean) deviation,
  variance, and standard deviation
• The idea is the figure out what is happening with
  your group

3
Percentiles

• Percentiles can get you started in helping you
  figure out the trends in your databut they have
  limitations
• Percentilea point in a distribution at or below
  which a given percentage of scores are found
• 49th percentile means that 49 of the scores fall
  at or below that particular point

4
Central Tendency

• Statisticians need a way to indicate where, along
  a scale of measures, a given distribution is
  centered.
• They used three measures of central tendency
• The Mode
• The Median
• The Mean

5
SPSS Frequencies

• Here is a sample of an SPSS printout providing
  you with measures of central tendency and
  variation

6
The Mode

• Consider the following data set
• The mode is the most frequent score. In this
  case, the most frequent score is 7

7
7
Mode and nominal data

• Mode is best measure of central tendency when
  data are not orderedlike the colors of cars in a
  parking lot.
• For colors of cars
• You cant use the medianthere is no order in the
  colors, no counting up from the bottom to find
  the middle score
• Also, you cannot add them together to find a
  mean.
• (blue red white?)
• Summary Mode is the place where the greatest
  number of cases, observations, scores occur

8
The Median

• Is the middle score.
• is the 50th percentile.
• The point on the scale of measurement below which
  50 of the scores fall.
• Steps for finding the median
• Arrange scores in ascending order
• If the number of scores is odd, the median is the
  middle score
• If the number of scores is even, the median is
  halfway between the two middle scores

9
Example of finding the median

• Consider this distribution 1, 2, 3, 4, 5,
• Its an odd number so the median is 3
• Consider this distribution 1, 2, 3, 4, 4, 5
• Its an even number so the median is halfway
  between the two middle scores, 3 and 4. The
  median is 3.5

10
Median Divides a Distribution in Half

• That point between the lower and upper halves of
  the distribution.
• It is not sensitive to the exact location of
  every score in the distribution, but is used as
  the measure of central tendency when the mean is
  inappropriate.
• It is used instead of the mean when the
  distribution is very skewed
• If distribution is symmetrical (see additional
  notes) mean, median, and mode are at the same
  point.
• If the distribution is skewed, the mean changes
  while the median does not.

11
Median example

• Consider the difference between mean faculty
  salaries at Texas Tech
• With head sports coaches salaries
• Without head sports coaches salaries
• In this case, the median would provide a better
  indication of the middle value of the
  distribution of faculty salaries
• A fix coaching salaries are so different that
  they deserve to be in another group
• Median is best measure of central tendency for
  ranked data too (e.g. order of runners in a race)
  
• Use the median if data are severely skewed

12
The mean

• Is the average of the scores in a distribution
• X-bar is the mean
• X-sub i is a given observation/score
• n is the number of scores
• (Note the formula above is for the sample mean.
  Nearly all of our calculations using the mean
  will use this formula.)

13
Properties of the Mean

• Sum of deviations of all scores from the mean is
  zero
• The sum of squares of the deviation from the mean
  is smaller than the sum of squares of the
  deviations from any other value in the
  distribution

14
Here is an example an Example
15
The Means Mu and X-bar

• If you have access to the entire population the
  formula for the mean is
• Where mu is the mean of the population

16
Just X-bar

• Note the formula for X-bar is
• Essentially the same as the formula for mu.
• The Roman X and the lower case n indicate that
  this is the formula for sample means.

17
Mean as Fulcrum of a Distribution.

• Imagine that you have a long, rigid beam that has
  14 wooden blocks of equal weight on it. At what
  point on the beam would a fulcrum have to be
  placed to establish equilibrium? What is the
  balance point of the distribution? The mean is
  that balance point of the distribution.
• Therefore the mean is the only measure of
  central tendency that is sensitive to all its
  scores. (See additional notes 1).
• This makes the mean important to calculating many
  types of statistics including standard deviation,
  product moment correlation, all types of standard
  errors, etc.

18
More on the Mean

• Very useful when making inferences from a sample
  to a population
• Mean of sample is best estimate of mean of
  population
• Of course, a sample mean is only an estimate.

19
Central tendency and Skewness

• Mean is pulled
• toward extreme
• Values
• Mode is at the
• peak of dist.
• Median is between
• mean and mode

20
Heres an SPSS histogram of the data set
5,7,3,38,7
21
Variability

• Different samples and populations will have
  different means.
• How scores vary from the mean will differ also.
• Some distributions will be flatthe scores are
  very spread out
• Some will present the scores squeezed together
• So the mean could be the same for two
  distributions and the variance could be very
  different.
• Helps to have a numerical index of how scores vary

22
Variability example

• Compare these two distributions. Each has the
  same mean and same n (number of observations).
  However, they are very different distributions
  because of how the scores or spread. Test1 is
  much more bunched that Test2. Test2 has more
  variability in it s scores.

23
Numerical indexes of variance

• We are interested in how scores vary from the
  mean (that all important central point)
• ADaverage deviation of scores from the mean (aka
  mean deviation). Limited utility with other
  statistics because of the use of absolute value.
  This statistic is rarely used
• Varianceaverage of the sum of squared deviations
  around the mean
• Standard deviationsquare root of the variance.
  It is important because it is expressed in the
  same units as the original data

24
Formulas for variability

• Note the similarity of these formulas
• Average deviation
• Variance
• Standard deviation

25
Compare Variability to Mean

• Note that both are averages. Variance is an
  average of the deviations from the mean. The
  mean is an average of the scores. The standard
  deviation is the variance expressed in linear,
  rather than squared units.

26
Other Measures of Variability

• IRQinterquartile rangethe score at Q3 the
  score at Q1
• IRQ is to the measures of variability what the
  median is to measures of central tendency.
• Preferred to standard deviation when distribution
  is radically asymmetrical
• Perfect companion to the median. If median is
  reported IRQ should be reported too.

27
The Range

• Range is the the difference between the highest
  score and the lowest score
• Lots of instability very sensitive to extremes
  scores
• It is dependent on only two scores
• But, quick to calculate