Week 1: Describing Data using Numerical Measures

1
Week 1 Describing Data using Numerical Measures
GOALS / LEARNING OUTCOMES When you have completed
this topic, you will be able to
ONECalculate the arithmetic mean, median, and
mode
TWO Explain the characteristics, uses,
advantages, and disadvantages of each measure of
location.
THREEIdentify the position of the arithmetic
mean, median, and mode for both a symmetrical and
a skewed distribution.
2
Week 1 Describing Data using Numerical Measures
FOUR Compute and interpret the range, the mean
deviation, the variance, and the standard
deviation of ungrouped data.
FIVEExplain the characteristics, uses,
advantages, and disadvantages of each measure of
dispersion.

3
Essential Reading

• Lind, Marchal and Wathen, (2003) Basic Statistics
  for Business and Economics, chapter 3

4
Characteristics of the Mean

• The arithmetic mean is the most widely used
  measure of location.

It is calculated by summing the values and
dividing by the number of values.
5
Population Mean

• The population mean is the sum of all the
  population values divided by the total number of
  values

where µ (pronounced mu) is the population mean. N
is the total number of observations. X is a
particular value. ? (sigma) indicates the
operation of adding.
6
EXAMPLE 1

• The Smith family owns four cars. The following
  is the current mileage on each of the four cars
• 56,000, 23,000,42,000,73,000
• Find the mean.
• 56,000, 23,000, 42,000, 73,000

7
Sample Mean

• The sample mean is the sum of all the sample
  values divided by the number of sample values
• Where n is the total number of values in the
  sample.

8
EXAMPLE 2

• A sample of five executives received the
  following bonus last year (000)
• 14.0, 15.0, 17.0, 16.0, 15.0

9
Properties of the Arithmetic Mean

• Every set of interval-level and ratio-level
  data has a mean.

• All the values are included in computing the
  mean.
• A set of data has a unique mean.
• The mean is affected by unusually large or small
  data values.
• The arithmetic mean is the only measure of
  central tendency where the sum of the deviations
  of each value from the mean is zero.

10
EXAMPLE 3

• Consider the set of values 3, 8, and 4. The
  mean is 5. Illustrating the fifth property

11
Weighted Mean

• The weighted mean of a set of numbers X1, X2,
  ..., Xn, with corresponding weights w1, w2,
  ...,wn, is computed from the following formula

12
EXAMPLE 4

• During a one hour period on a hot Saturday
  afternoon Chris served fifty drinks at a kiosk.
  He sold five drinks for 0.50, fifteen for 0.75,
  fifteen for 0.90, and fifteen for 1.15.
  Compute the weighted mean of the price of the
  drinks.

13
The Median

• The Median is the midpoint of the values after
  they have been ordered from the smallest to the
  largest.

• There are as many values above the median as
  below it in the data array.
• For an even set of values, the median will be
  the arithmetic average of the two middle numbers.

14
EXAMPLE 5

• The ages for a sample of five university
  students are
• 21, 25, 19, 20, 22

Arranging the data in ascending order gives 19,
20, 21, 22, 25. Thus the median is 21.
15
Example 6

• The heights of four basketball players, in
  centimetres, are
• 176, 173, 180, 175

• Arranging the data in ascending order gives 173,
  175, 176, 180. Thus the median is 175.5, found
  by (175176)/2.

16
Properties of the Median

• There is a unique median for each data set.

• It is not affected by extremely large or small
  values and is therefore a valuable measure of
  central tendency when such values occur.
• It can be computed for ratio-level,
  interval-level, and ordinal-level data.

17
The Mode

• The mode is the value of the observation that
  appears most frequently.

• EXAMPLE 7 The exam scores for ten students are
  81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because
  the score of 81 occurs the most often, it is the
  mode.

18
Geometric Mean

• The geometric mean (GM) of a set of n numbers is
  defined as the nth root of the product of the n
  numbers. The formula is
• 
  
  
  

The geometric mean is used to average percentage
values and index values.
19
EXAMPLE 8

• The interest rate on three bonds were 5, 41, and
  4 percent.
• The geometric mean is

20
Example 8 continued
The arithmetic mean is (1.051.411.04)/3 1.1667

The GM gives a more conservative profit figure
because it is not heavily weighted by the rate of
41percent.
21
Range

• The range is the difference between the largest
  and the smallest value.

• Only two values are used in its calculation.
• It is influenced by an extreme value.
• It is easy to compute and understand.

22
Mean Deviation

• The Mean Deviation is the arithmetic mean of the
  absolute values of the deviations from the
  arithmetic mean.

23
The Mean Deviation

• The main features of the mean deviation are

• All values are used in the calculation.
• It is not unduly influenced by large or small
  values.
• The absolute values are difficult to manipulate.

24
Mean Deviation

• The formula is

25
EXAMPLE 10

• The weights of a sample of crates containing
  books for the bookshop (in kilogrammes) are
• 103, 97, 101, 106, 103
• Find the range and the mean deviation.

• Range 106 97 9

26
Example 11

• To find the mean deviation, first find the mean
  weight.

27
Example 11 continued

• The mean deviation is

28
Population Variance

• The population variance is the arithmetic mean
  of the squared deviations from the population
  mean.

29
Population Variance

• The major characteristics are

• All values are used in the calculation.
• Not influenced by extreme values.
• The units are awkward, the square of the original
  units.

30
Variance

• The formula for the population variance is

31
Variance

• The formula for the sample variance is

32
EXAMPLE 12

• The ages of the Dunn family are
• 2, 18, 34, 42
• What is the population variance?

33
The Population Standard Deviation

• The population standard deviation s is the square
  root of the population variance.
• For EXAMPLE 12, the population standard deviation
  is 15.19, found by

34
EXAMPLE 13

• The hourly wages earned by a sample of five
  workers are
• 7, 5, 11, 8, 6.
• Find the variance.

35
Sample Standard Deviation

• The sample standard deviation is the square root
  of the sample variance.

In EXAMPLE 13, the sample standard deviation is
2.30
36
Interpretation and Uses of the Standard Deviation

• Empirical Rule For any symmetrical, bell-shaped
  distribution

• About 68 of the observations will lie within 1
  standard deviation of the mean,

• About 95 of the observations will lie within 2
  standard deviations of the mean

• Virtually all the observations will be within 3
  standard deviations of the mean

37
m-2s
m-1s
m
m1s
m2s
m 3s
m-3s