This document discusses measures of central tendency and dispersion used in descriptive statistics. It defines the mode, median, and mean as measures of central tendency and how to calculate each for both raw and grouped data. It also discusses properties of each measure. For measures of dispersion, it defines the range, interquartile range, variance and standard deviation, providing formulas and examples of calculating each for a set of test score data. It concludes with notes on rounding rules and homework questions.

1. Descriptive Statistics
by Indramani Tripathi
Measures of Central Tendency
And Dispersion

2. Measures of Central Tendency
 1. Mode = can be used for any kind of data
but only measure of central tendency for
nominal or qualitative data.
 Formula: value that occurs most often or the
category or interval with highest frequency.
 Note: Omit Formula 3.1 Variation Ratio in
Healey and Prus 2nd
Cdn.

3. Example for Nominal Variables:

 Religion frequency cf proportion % Cum%
 Catholic 17 17 .41 41 41
 Protestant 4 21 .10 10 51
 Jewish 2 23 .05 5 56
 Muslim 1 24 .02 2 58
 Other 9 33 .22 9 80
 None 8 41 .20 20 100

 Total 41 1.00 100%
 Central Tendency: MODE = largest category = Catholic

4. Central Tendency (cont.)
 2. Median = exact centre or middle of
ordered data. The 50th percentile.
 Formula:
 Array data.
 When sample size is even, median falls
halfway between two middle numbers.
 To calculate: find (n/2) and (n/2)+1, and
divide the total by 2 to find the exact median.
 When sample size is odd, median is exact
middle (n+1) /2

5. Example for Raw Data:
 Suppose you have the following set of test
scores:
 66, 89, 41, 98, 76, 77, 68, 60, 60, 67, 69, 66,
98, 52, 74, 66, 89, 95, 66, 69
 1. Array (put in order) your data:
 98 98 95 89 89 77 76 74 69 69
68 67 66 66 66 66 60 60 52 41
N = 20 (N is even)

6. To calculate:
- find middle numbers(n/2)+(n/2 )+1
- add together the two middle numbers
- divide the total by 2
 First middle number: (20/2) = the 10th
number
 2nd
middle number: (20/2)+1 = the 11th
number
 Look at data:
the middle numbers are 69 and 68
 The median would be (69+68)/2 = 68.5

7. Median for Aggregate (grouped) Data
 This formula is shown in Healey 1st
Cdn
Edition but NOT in 2/3 Cdn
 We will NOT COVER this one!

8. Properties of median:
 - for numerical data at interval or ordinal level
 -"balance point“
 -not affected by outliers
 -median is appropriate when distribution is
highly skewed.

9. 3. Mean for Raw Data
 The mean is the sum of measurements /
number of subjects
 Formula: (X-bar) = ΣXi / N
 Data (from above):
66, 89, 41, 98, 76, 77, 68, 60, 60, 67, 69, 66,
98, 52, 74, 66, 89, 95, 66, 69

10. Example for Mean
 Formula: = ΣXi / N
= 1446 / 20
= 72.3
The mean for these test scores is 72.30

11. Mean for Aggregate (Grouped) Data
(Note: not in text but covered in class)
 To calculate the mean for grouped data, you
need a frequency table that includes a
column for the midpoints, for the product of
the frequencies times the midpoints (fm).
Formula: = Σ (fm)
N

12. Frequency table:
Score f m* (fm)
41-50 1 45.5 45.5
51-60 3 55.5 166.5
61-70 8 65.5 524
71-80 3 75.5 226.5
81-90 2 85.5 171
91-100 3 95.5 286.5
N = 20 Σ (fm) = 1420
* Find midpoints first

13. Calculating Mean for Grouped Data:
Formula: = Σ (fm)
N
= 1420 / 20
= 71
The mean for the grouped data is 71.

14. Properties of the Mean:
- only for numerical data at interval level
- "balance point“
- can be affected by outliers = skewed distribution
- tail becomes elongated and the mean is pulled in
direction of outlier.
Example…
no outlier:
$30000, 30000, 35000, 25000, 30000 then mean = $30000
but if outlier is present, then:
$130000, 30000, 35000, 25000, 30000 then mean = $50000
(the mean is pulled up or down in the direction of the outlier)

15. NOTE:
 When distribution is symmetric,
mean = median = mode
 For skewed, mean will lie in direction of skew.
 i.e. skewed to right (tail pulled to right)
mean > median (positive skew)
 skewed to left (tail pulled to left)
median > mean (negative skew)

16. Measures of Dispersion
 Describe how variable the data are.
 i.e. how spread out around the mean
 Also called measures of variation or
variability

17. Variability for Non-numerical Data
(Nominal or Ordinal Level Data)
 Measures of variability for non-numerical
nominal or ordinal) data are rarely used
 We will not be covering these in class
 Omit Formula 4.1 IQV in Healey and Prus
1st
Canadian Edition
 Omit Formula 3.1 Variation Ratio in Healey
and Prus 2/3 Canadian Edition

18. 2. Range (for numerical data)
Range = difference between largest and
smallest observations
i.e. if data are $130000, 35000, 30000, 30000,
30000, 30000, 25000, 25000
then range = 130000 - 25000 = $105000

19. Interquartile Range (Q):
- This is the difference between the 75th and the 25th
percentiles (the middle 50%)
- Gives better idea than range of what the middle of
the distribution looks like.
Formula: Q = Q3 - Q1 (where Q3 = N x .75,
and Q1 = N x .25)
Using above data: Q = Q3 - Q1 = (6th
– 2nd
case)
= $30000-25000 =$5000
The interquartile range (Q) is $5000.

20. 3. Variance and Standard Deviation:
 For raw data at the interval/ratio level.
 Most common measure of variation.
 The numerator in the formula is known as
the sum of squares, and the denominator is
either the population size N or the sample
size n-1
 The variance is denoted by S2
and the
standard deviation, which is the square root
of the variance, by S

21. Definitional Formula for Variance and
Standard Deviation:
 Variance: s2
= Σ (xi - )2
/ N
 Standard Deviation:
s =
 (the standard deviation is the square root of
the variance; the variance is simply the
standard deviation squared)

22. Example for S and S2
:
 Data: 66, 89, 41, 98, 76, 77, 68, 60, 60, 67,
69, 66, 98, 52, 74, 66, 89, 95, 66, 69
1. Find ∑ Xi
2
: Square each Xi and find total.
2. Find (∑ Xi)2
: Find total of all Xi and square.
3. Substitute above and N into formula for S.
4. For S2
, simply square S.
S = 14.75 S2
= 217.71

23. A working formula for the standard
deviation:
Note: the definitional formula for standard deviation is
not practical for use with data when N>10.
The working formula, which is much easier to do on
your calculator, should be used instead.
Both formulae give exactly the same result. Try it!
2
2
X
N
X
S i
−=
∑

24. Properties of S:
 always greater than or equal to 0
 the greater the variation about mean,
the greater S is
 n-1 corrects for bias when using sample data. S
tends to underestimate the real population standard
deviation when based on sample data so to correct
for this, we use n-1. The larger the sample size, the
smaller difference this correction makes. When
calculating the standard deviation for the whole
population, use N in the denominator.

25. NOTE:
 σ, N and Mu (µ) denote population
parameters
 s, n, x-bar ( ) denote sample statistics

26. Remember the Rounding Rules!
 Always use as many decimal places as your
calculator can handle.
 Round your final answer to 2 decimal places,
rounding to nearest number.
 Engineers Rule: When last digit is exactly 5
(followed by 0’s), round the digit before the
last digit to nearest EVEN number.

27. Homework Questions
 Healey and Prus 1e:
 #3.1, #3.5, #3.11 and 4.9, #4.15
 Healey and Prus 2/3e
 #3.1, #3.5, #3.11 (compute s for 8 nations also), #3.15
 SPSS:
 Read the SPSS sections for Ch. 3 and 4 in 1st
Cdn. Edition
and for Ch. 4 in 2/3 Cdn. Edition
 Try some of the SPSS exercises for practice