The interview, ethics and analysis

1.0 Introduction 2.0 Objectives 3.0 Main Content

3.1 Mean 3.2 Median 3.3 Mode 4.0 Conclusion 5.0 Summary

6.0 Tutor-Marked Assignment 7.0 References/Further Reading 1.0 INTRODUCTION

Measures of central tendency are also known as averages. These generally indicate where the centre of the data spread lies. The most popular and frequently used are; Mean, Median and Mode. However each has its peculiar use and application and none could be of much use and application without relating it the measure of variability of the population spread such as standard deviation or coefficient of variation.

2.0 OBJECTIVES

At the end of this Unit, you should be able to;

· define and understand the use of mean as a measure of central tendency

· know and understand the various types of mean

· define and understand the use of median as a measure of central tendency

· define and understand the use of mode as measures of central tendency

· know how to calculate the mean, median and mode

· understand the advantages and limitations in the use mean, median and mode as measures of central tendency

3.0 MAIN CONTENT 3.1 The Mean

· Arithmetic Mean

This measure implies arithmetic average or mean which is commonly denoted as i.e. pronounced as -bar. This is the sum of all observations divided by the total number of observations in an ungrouped data and given as;

̅ =

……..

The Arithmetic mean in table 1 will be

̅ =

…….. = 62.52

In grouped data it is given as; =

where Σ is the summation of all the groups, f is the class frequency, is the midpoint value in each group and n the total number of observations.

Characteristics and uses of Arithmetic Mean

· It is the centre of gravity of all values.

· In calculating arithmetic mean all observations are used.

· It is a stable average. Though affected by all values it is the least affected of all averages by fluctuations in samples.

· Sample means show less variation than individual values.

· It is not necessarily a value belonging to the group.

· Cannot be used when handling qualitative variables.

· Cannot be calculated if a single observation is missing.

· Cannot be calculated if extreme class is open i.e. no lower or upper limit e.g. < 20 or >50 class.

· Not a preferred average in skewed distribution.

· Variability of sample means depends on sample size.

· Its value is statistically useful when attached to standard deviation which may vary as shown in Fig. 13.

· It bisects a normal distribution into two equal halves.

Mean and standard deviation in a normal distribution

σA

σB

µ

Fig. 13. Normal distribution curve

The two distributions have the same mean ‘µ’ but different standard deviations σA and σBi.e. the values measured in population B are more scattered and spread than in population A. So comparing their means without considering their standard deviations will be misleading.

· Weighted mean ( w)

In certain situations not all the observations will be measured precisely as others. Therefore a rational way out is to give relatively more weight to the more precise observations. Thus if the observations x1, x2, x3...,xn have associated weights w1, w2, x3,...,wn respectively then;

w = ^………,

……….,

· Harmonic mean ( H)

Thisis the average of all the arithmetic means of various populations or groups of measurement. If the various arithmetic means of various groups are given as 1, 2, 3,... n, then;

H = ^……… or H =

· Geometric Mean ( g)

Thisis the nth root of the product of all observations

g = … … .

Characteristics and uses of Geometric Mean

· Geometric mean is zero when any observation is zero.

· Geometric mean is imaginary when any observation is negative irrespective of the magnitude of other observations.

· Geometric mean is used in calculating population growth rate.

3.2 Median

The median is the middle most value in a set of ordered observations that ends with an odd number. The average of the two most middle values is taken when the set of the ordered values end with an even number i.e.

· When n is odd, Median = _{( )/}

· When n is even, Median = ½[ _{( )}+ χ_{( )}]

· When the data is grouped, Median = L + where L is the value of the lower boundary of the group where the median falls, n is the total observations, F is the total frequency below the group where the median falls, ℓ is the class interval and is the class frequency of the group where the median falls.

Characteristics and uses of the Median

· It is used in irregular and skewed distributions.

· It is not affected by outliers.

· Can be calculated if extreme class is open i.e. no lower or upper limit e.g. < 20 or >50 class.

· Can be used while dealing with qualitative variables.

· It is a better index for describing average number of cases of communicable diseases which have well defined cycle.

· It is also a better average in describing the trend of an illness over a period of time including epidemic years.

· Cannot be determined directly for even number observations.

· It is not based on all observations.

3.3 The Mode

Mode is the value that has the highest frequency i.e. most occurring value in a set of values. In a discrete probability distribution it is the value at which its probability mass function takes its maximum value or at the peak i.e. the value that is most likely to be sampled.

· Types of Mode

The distribution is unimodal if only one value has the highest frequency and bimodal if two values. For more than one value with highest frequencies, it is considered multimodal.

Fig. 14a. Unimodal distribution. Fig. 14b. Bimodal distribution Characteristics and uses of the Mode

· Does not exist if all values in a set occur with the same frequency.

· Not affected by outliers.

· It gives the point at which the observations cluster and converge.

· May have multiple values (mode).

· Not based on all observations.

· Used when the most repeated variable is wanted.

· Affected by fluctuation of sampling.

4.0 CONCLUSION

In this unit we discussed that measures of central tendency are also known as averages that generally indicate where the centre of the data spread lies. It was explained that the most popular and frequently used measures of central tendency are the Mean, Median and Mode. It was noted, however each has its peculiar use and application and none could be of much use and application without relating it the measure of variability of the population spread such as standard deviation or coefficient of variation.

5.0 SUMMARY In this unit we learnt;

· how to define and understood the uses of mean as a measure of central tendency

· the various types of mean as used as measure of central tendency

· the use of median as a measure of central tendency

· the use of mode as measures of central tendency

· how to calculate the mean, median and mode

· the applications, advantages and limitations in the use of mean, median and mode as measures of central tendency

6.0 TUTOR-MARKED ASSIGNMENT

1. Define and explain the use of mean as a measure of central tendency.

2. Mention the various types of mean as used as measure of central tendency.

3. Define and explain the use of median as a measure of central tendency.

4. Define and explain the use of mode as measures of central tendency.

5. What are the various types of Mode?

6. Explain how to calculate the mean, median and mode.

7. What are the applications, advantages and limitations in the use of mean, median and mode as measures of central tendency?

7.0 REFERENCES/FURTHER READING

Mahajan, B. K. (2010). Methods in Biostatistics for Medical Students and Research Workers.7^th edition. Jaypee Brothers Medical Publishers Ltd. New Delhi: 33-52.

Murray, R. S and Larry, J. S. (2006). Statistics. Theory and Problems of Statistics. 3^rd edition. Tata McGraw-Hill. New Delhi: 1-7.

Ogbonna, C. (2016). The Basics in Biostatistics, Medical Informatics and Research Methodology. 3 in 1 Book.Revised Ed. Yakson Printing Press. Jos: 3-128.

Petrie, A. (1986). Lecture Notes on Medical Statistics. Blackwell Scientific Publications Ltd. Edinburgh: 40-46.

Singha, P. (1996). An Introductory Text on Biostatistics. 2^nd edition.

Habason Nig. Limited. Kano: 32-34.

Syvia, Wassertheil-Smoller. (1990). Biostatistics and Epidemiology. A primer for health professionals. Springer-Verlag. New York:

119.

Wayne, W. D. (2006). Biostatistics. A Foundation for Analysis in the Health Sciences. 7^th edition. John Wiley and Sons. New Delhi:

57-71.

UNIT 2 MEASURES OF LOCATION

In document A proposed model for servitization based collaboration in the UK Aerospace Defence industry (Page 111-121)