Measures of Central Tendency

Measures of Central Tendency ◦ Also known as measures of central location (locate central distribution). ◦ “Three kinds of averages of a data set” to answer “where do the data center?” ◦ Measures include: 1.

2.

3.

Mean Mode Median

Pharmaceutical Biostatistics: Descriptive Statistics

The Mean ◦ The usual “average” that is familiar to everyone. ◦ Adds up all the numbers (Σ x) and divide by how many numbers there are (N for population or n for sample). ◦ Formula: Sample mean : Population mean :

Pharmaceutical Biostatistics: Descriptive Statistics

The Mean n 

Example: The reduction in blood pressure (mmHg) in 6 patients 4 hours after administration of a standard dose of a novel antihypertensive agent is shown in Table 1.1.

Calculate the mean reduction in blood pressure reduction in the 6 patients.

Table 1.1 Effect of an antihypertensive drug on blood pressure lowering in six patients

Patient number

Reduction in blood pressure (mmHg)

1

20

2

25

3

21

4

34

5

41

6

37 Pharmaceutical Biostatistics: Descriptive Statistics

The Mean n Substituting

the figures from Table 1.1 into the equation for the mean, we obtain: = (20+25+21+34+41+37)/6 =178/6 =29.67 mmHg

Pharmaceutical Biostatistics: Descriptive Statistics

The Weighted (arithmetic) Mean ◦ Each datum point in the distribution does not contribute equally to the overall calculation of the mean. ◦ Data is divided into groups, each of which possesses different weighting. ◦ Formula:

Pharmaceutical Biostatistics: Descriptive Statistics

The Weighted (arithmetic) Mean n 

Example: The effect of a defined dose of a commercially available analgesic to suppress pain following a painful stimulus was evaluated in 20 volunteers using an analogue scale (Table 1.2). Calculate the mean of the pain assessment by the 20 volunteers.

Table 1.2 Recorded assessment of pain by 20 volunteers following administration of analgesic and exposure to a painful stimulus

Number of volunteers

Pain assessment by volunteers

2

3 (extreme pain)

12

2 (moderate pain)

6

1 (slight pain) Pharmaceutical Biostatistics: Descriptive Statistics

The Weighted (arithmetic) Mean n Substituting

the figures from Table 1.2 into the equation for the weighted arithmetic mean, we obtain: = (2x3)+(12x2)+(6x1)/20 =36/20 =1.8

Pharmaceutical Biostatistics: Descriptive Statistics

The Weighted Mean (Frequency Distribution) Diameter (mm)

Frequency

Midpoint (x)

f.x

35-39

6

37

222

40-44

12

42

504

45-49

15

47

705

50-54

10

52

520

55-59

7

57

399

Total

50

2350

Mean=2350/50 =47

Pharmaceutical Biostatistics: Descriptive Statistics

The Median ◦ An alternative method of describing the central nature of data. ◦ Relatively unaffected by the nature of the spread of data. ◦ Is the middle number. It is found by putting the numbers in order and taking the actual middle number if there is one, or the average of the two middle numbers if not.

Pharmaceutical Biostatistics: Descriptive Statistics

The Median ◦ Consider the following data: A random samples of yearly income of 7 employees (rounded to the nearest hundred dollars) 24.8 22.8 24.6 192.5 25.2 18.5 23.7 a) The mean (rounded in 1 decimal place is) : 47.4, but the statement “the average income of 7 employees is $47, 400” is certainly misleading.

Pharmaceutical Biostatistics: Descriptive Statistics

The Median – outliers 24.8 22.8 24.6 192.5 25.2 18.5 23.7 ◦ Number 192.5 is called outliers (far removed from most or all the remaining measurements). *mean is sensitive to extreme values ◦ Usually is the result of some sort of error (but not always). ◦ So, a better measure of the “center” of the data can be obtained if we were to arrange the data in numerical order.

Pharmaceutical Biostatistics: Descriptive Statistics

The Median ◌ The order 18.5 22.8 23.7 24.6 24.8 25.2 192.5 Then select the middle number in the list, in this case 24.6.

In this sense, it locates the center of the data.

Pharmaceutical Biostatistics: Descriptive Statistics

The Median If there are an even number of measurements in the data sets, there will be two middle elements -> take the mean of middle two as the median Example: n=8 18.5 22.8 23.7 24.6 24.8 25.2 28.9 192.5 Median: (24.6+24.8)/2 = 24.7

Pharmaceutical Biostatistics: Descriptive Statistics

The Mode ◦ The easiest measure of the average. ◦ Defined as the item of data with the highest frequency. ◦ Most frequently occurring number. ◦ For any data set there is always exactly one mean and exactly one median. ◦ However, several different values could occur with the highest frequency.

Pharmaceutical Biostatistics: Descriptive Statistics

The Mode Data set:

-1 0 2 0

The mode of the following data set is 0.

Data set:

2 2 3 1 1 5

Two most frequently observed values in this data set are 1 and 2. Therefore mode is a set of two values : {1,2}

Pharmaceutical Biostatistics: Descriptive Statistics

Example (Mode, Median and Mean) Weight of luggage presented by airline passengers at check-in (measured to the nearest kg).

18 23 20 21 24 23 20 20 15 19 24 Mode: 20 (this number occurs 3 times).

Median: put the numbers in order first and take the actual middle number (odd count) or the average middle number (even count).

15 18 19 20 20 20 21 23 23 24 24 : 20 Mean: 15+18+19+20+20+20+21+23+23+24+24 / 11 : 20.64 Pharmaceutical Biostatistics: Descriptive Statistics

Choice of the mean or median to describe the central tendency n  For

normal distributed data, the numerical values of the mean and mean should be identical and either term may successfully be used to describe the central point.

n  The

use of median is preferable for distributions that possess extreme values (mean is unacceptably distorted).

Pharmaceutical Biostatistics: Descriptive Statistics

When not to use the mean? ◦ Mean is good for dataset that is evenly spread.

Staff

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Salary 15k

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18k

16k

14k

15k

15k

12k

17k

90k

95k

◦ The mean salary is 30.7k.

Pharmaceutical Biostatistics: Descriptive Statistics