Descriptive Statistics

1

• Descriptive Statistics
• Outline
• Measures of Central Tendency
• Mode
• Median
• Mean
• Measures of Variability
• Range
• Variance standard deviation

2
Measures of Central Tendency

• Measures of central tendency tell you what is
  true on average

• 3 such measures are used regularly
• Mode
• Median
• Mean

3
The Mode

• 5 6 6 6 7 7 8 9 10 10 10 11 12

• The most common value(s) in a data set.
• This is a bi-modal distribution (it has 2
  modes)

Mode 1 Mode 2
4
The Median

• The median of a data-set is the value in the
  middle
• That is, half of the scores in the set lie above
  and half lie below it

• 5 6 6 6 7 7 8 9 10 10 10 11 12

6 scores Median
5
The Mean

• The mean is the arithmetic average of a set of
  numbers.

• Most frequently used of the three measures
• Most useful when distribution is not skewed.

6
Note 1 Skewed distributions

• A skewed distribution is one in which the scores
  pile up at one end of the scale, but are less
  frequent at the other end

• 4 5 7 8 11 14
• (not skewed)
• 2 4 8 22 300 14000 (skewed)

Scores are rare among the larger numbers
7
Measures of Central Tendency

• We distinguish between SAMPLE and POPULATION
  values.

• Sample values are shown as English letters
• Population values are shown as Greek letters

8
Measures of Central Tendency

• A sample mean is the average of all values in
  the sample.
• Sample mean X
• This is pronounced X-bar

• A population mean is the average of all values
  in the population
• Population mean ? (Mu)

9
Note 2 Sigma notation

• The Greek letter ? indicates the addition
  operation.
• ?(xi) x1 x2 x3 x4
• i1

4
10
The Mean - calculations

• 4 5 7 8 11 14
• Sx 49 n 6
• X Sx 49 8.17
• n 6

• Sx ? The sum of X
• n 6 ? because there are 6 observations in the
  data set

X bar
11
The Mean - calculations

• 4 5 7 8 11 14
• Sx 49 n 6
• X Sx 49 8.17
• n 6

• 2 4 8 22 300 14000
• Sx 14336 n 6
• X Sx 14336
• n 6
• 2389.333

12
Measures of Variability

• With any sample mean, the question arises, how
  useful is this number to what extent is it
  descriptive of the data set?

• The answer depends upon how variable the data
  set is how similar each data point is to all
  the other data points in the set.

13
Measures of variability

• The range.
• The distance between the highest and lowest
  numbers in the data set
• Simplest and least useful measure of variability
  is the range

• 5 6 6 6 7 7 8 9 10 10 10 11 12
• Here, the range is 12 5 7

14
Measures of Variability

• The Variance
• measures how much on average each data point is
  different from the others.
• much more useful than the range

• To compute variance
• Find mean, X
• Subtract X from each data point Xi
• Square differences add squared values up
• Divide total by n-1

15
The variance

• Why do we square the differences before adding
  them up?
• Because if we didnt, the differences would
  always add up to zero.

• Sample variance S2
• (S-squared)
• Population variance s2 (sigma squared)
• Important for you to understand how S2 is
  different from s2.

16
Conceptual Formula for the Variance

• S2 ?(Xi X)2
• n-1

S2 is the sample variance because X is the sample
mean
17
Note degrees of freedom

• Suppose we have six scores, and we know that X
  8
• Suppose five of the six scores are
• 3, 5, 9, 10, 12

• What is the last score?

18
Note degrees of freedom

• Sum of the five scores
• 3591012 39
• X 8 so sum of all six scores is 6 8 48
• So, unknown score is 48 39 9

• There are n-1 degrees of freedom in n scores
• Given n 1 scores and the sample mean, there is
  no uncertainty about what the remaining score is
  (that score is not free to vary).

19
10 x 14.5 145 145 125 20 Thus, there are
nine degrees of freedom in ten observations
20
Computational Formula for the Variance

• S2 ?X2 (?X)2
• n
• (n-1)
• Note S ?S2

Note these are Xs, not X-bars