1.0 Introduction 2.0 Objectives 3.0 Main Content
3.1 Properties of Normal Distribution 3.2 Importance of Normal Distribution 4.0 Conclusion
5.0 Summary
6.0 Tutor-Marked Assignment 7.0 References/Further Reading 1.0 INTRODUCTION
A normal distribution also known as Gaussian curve is one that is unimodal with the total area under the curve 100% or a unit. It is said to be symmetrical about µ with Mean, Mode and Median equal and lie on the same axis. It is characterised by population mean, µ and variance, σ2 and for a constant σ a change in µ1 to µ2 shifts the curve along the x-axis to the right, if µ2>µ1 and to the left when µ1>µ2. For a constant µ a change in σ from σ1 to σ2 alters the peakedness of the curve. It is more peaked (taller or thinner) if σ1 > σ2 and less peaked (fatter or flatter) if σ2
> σ1. X-axis is an asymptote to curve while the intervals on either sides of µ encloses approximately a total probability of; 68.27% for 1 SD, 95.45% for 1.96 SD and 99.73% for 2.58 SD.
2.0 OBJECTIVES
At the end of this unit, you should be able to;
· define and explain the concept in Normal Distribution
· mention and understand the properties of Normal Distribution
· discuss the importance of Normal Distribution or Gaussian curve
· discuss the uses and applications of Normal Distribution population.
3.0 MAIN CONTENT
3.1 Properties of a Normal Distribution Curve
P (χ) =√ /( ) (- ∞ < χ < ∞)
Fig. 21. Normal distribution (Gaussian curve)
· It is bell shaped i.e. unimodal
· Total area under the curve is 1 (100%) or a unit.
· Symmetrical about µ i.e. can be bisected into two equal symmetric halves.
· Mean, Mode and Median coincide (equal) i.e. lie on the same axis.
· Characterised by population mean, µ and variance, σ2
· For a constant σ a change in µ1 to µ2 shifts the curve along the x-axis to the right, if µ2>µ1 and to the left when µ1>µ2.
· For a constant µ a change in σ from σ1 to σ2 alters the peakedness of the curve. It is more peaked (taller or thinner) if σ1 > σ2 and less peaked (fatter or flatter) if σ2 > σ1
· X-axis is an asymptote to curve.
· The intervals on either sides of µ encloses approximately a total probability of;
· 68.27% for 1SD
· 95.45% for 1.96SD
· 99.73% for 2.58SD
Areas covered by each standard deviation about the mean are shown below.
-2.58 -1.96 -1 µ+1 +1.96 +2.58
Fig. 22. Probability intervals under the normal distribution curve 3.2 Importance of Gaussian curve
· If variation is produced by large number of effects the distribution is normal.
· It fits most practical distribution occurring in real life and medicine i.e. Binomial and Poisson (P) can be approximated by normal distribution, provided that P is neither too close to zero nor unit and n is large.
· When the distribution is not normal, transformation techniques exist to make it normal.
· Sampling distributions of means and proportions are known to have normal distribution.
· Many distributions of sample statistic tend to normal for large samples and as such they can be studied with the help of normal distribution.
· If variables are not normally distributed, transformation techniques to make them normal exist.
· The entire theory of small sample tests such as, t, F and X2 tests is based on the fundamental assumption that the parent population from which the sample is drawn follows a normal distribution.
· The statistical theory is elegant if assumptions hold.
· It is the cornerstone of all parametric tests of statistical significance.
4.0 CONCLUSION
In this unit we learnt that a Normal Distribution is also known as Gaussian curve. It is one that is unimodal with the total area under the curve 100% or a unit. The other features are symmetry with the Mean, Mode and Median equal and lie on the same axis. The population distribution is characterized by mean and variance and for a constant variance. Changing the Mean value shifts the curve along the x-axis.
When the Mean is constant a change in Standard deviation will alter the peakedness of the curve. That X-axis is an asymptote to the curve while the intervals on either sides of the Mean will enclose approximately a total probability of; 68.27% for 1 SD, 95.45% for 1.96 SD and 99.73%
for 2.58 SD respectively.
5.0 SUMMARY
In this unit you have learnt:
· the definition of Normal Distribution and can explain the concept
· the properties of a Normal Distribution
· the importance of Normal Distribution
· the uses and applications of Normal Distribution population
· proportion of values covered on both sides of the mean for a given standard deviation.
6.0 TUTOR-MARKED ASSIGNMENT
1. Define Normal Distribution and can explain the concept.
2. List and explain the properties of a Normal Distribution.
3. What is the importance of Normal Distribution?
4. What are uses and applications of Normal Distribution population?
5. Mention the percentage values covered on both sides of the mean for a given standard deviation.
7.0 REFERENCES/FURTHER READING
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. 2nd edition.
Habason Nig. Limited. Kano: 32-34.
Syvia, Wassertheil-Smoller. (1990). Biostatistics and Epidemiology. A primer for health professionals. Springer-Verlag. New York:
119.
UNIT 2 STANDARD NORMAL DISTRIBUTION