NORMAL DISTRIBUTION AND ITS APPL ICATION

INTRODUCTION Statistically, a population is the set of all possible values of a variable. Random selection of objects of the population makes the variable a random variable ( it involves chance mechanism) Example: Let ‘x’ be the weight of a newly born baby. ‘x’ is a random variable representing the weight of the baby. The weight of a particular baby is not known until he/she is born.

Discrete random variable: If a random variable can only take values that are whole numbers, it is called a discrete random variable. If a random variable can only take values that are whole numbers, it is called a discrete random variable. Example: No. of daily admissions No. of boys in a family of 5 No. of boys in a family of 5 No. of smokers in a group of 100 No. of smokers in a group of 100 persons. persons. Continuous random variable: If a random variable can take any value, it is called a continuous random variable. Example: Weight, Height, Age & BP.

Continuous Probability Distributions Continuous distribution has an infinite number of values between any two values assumed by the continuous variable Continuous distribution has an infinite number of values between any two values assumed by the continuous variable As with other probability distributions, the total area under the curve equals 1 As with other probability distributions, the total area under the curve equals 1 Relative frequency (probability) of occurrence of values between any two points on the x-axis is equal to the total area bounded by the curve, the x-axis, and perpendicular lines erected at the two points on the x-axis Relative frequency (probability) of occurrence of values between any two points on the x-axis is equal to the total area bounded by the curve, the x-axis, and perpendicular lines erected at the two points on the x-axis

The Normal or Gaussian distribution is the most important continuous probability distribution in statistics. The term “Gaussian” refers to ‘Carl Freidrich Gauss’ who develop this distribution. The word ‘normal’ here does not mean ‘ordinary’ or ‘common’ nor does it mean ‘disease-free’. It simply means that the distribution conforms to a certain formula and shape.

Histograms A kind of bar or line chart A kind of bar or line chart Values on the x-axis (horizontal) Values on the x-axis (horizontal) Numbers on the y-axis (vertical) Numbers on the y-axis (vertical) Normal distribution is defined by a particular shape Normal distribution is defined by a particular shape Symmetrical Symmetrical Bell-shaped Bell-shaped

Histogram Figure 1 Histogram of ages of 60 subjects

A Perfect Normal Distribution

Gaussian Distribution Many biologic variables follow this pattern Many biologic variables follow this pattern Hemoglobin, Cholesterol, Serum Electrolytes, Blood pressures, age, weight, height Hemoglobin, Cholesterol, Serum Electrolytes, Blood pressures, age, weight, height One can use this information to define what is normal and what is extreme One can use this information to define what is normal and what is extreme In clinical medicine 95% or 2 Standard deviations around the mean is normal In clinical medicine 95% or 2 Standard deviations around the mean is normal Clinically, 5% of “normal” individuals are labeled as extreme/abnormal Clinically, 5% of “normal” individuals are labeled as extreme/abnormal We just accept this and move on. We just accept this and move on.

Normal distribution  Most important distribution in statistics  Also called the Gaussian distribution  Density given by  for -  < x <   where  is the mean and  the standard deviation

Area under a smooth curve Integration of the density function over the range a to b Integration of the density function over the range a to b Density function is a formula used to represent the distribution of a continuous random variable Density function is a formula used to represent the distribution of a continuous random variable A nonnegative function f(x) is called a probability distribution or probability density function of the continuous random variable X if the total area bounded by its curve and the x- axis is equal 1 and if the sub area under the curve bounded by the curve, the x-axis, and perpendiculars erected at any two points a and b gives the probability that X is between the points a and b A nonnegative function f(x) is called a probability distribution or probability density function of the continuous random variable X if the total area bounded by its curve and the x- axis is equal 1 and if the sub area under the curve bounded by the curve, the x-axis, and perpendiculars erected at any two points a and b gives the probability that X is between the points a and b

Gaussian or Normal Distribution Curve

Characteristics of Normal Distribution Symmetrical about mean,  Symmetrical about mean,  Mean, median, and mode are equal Mean, median, and mode are equal Total area under the curve above the x- axis is one square unit Total area under the curve above the x- axis is one square unit 1 standard deviation on both sides of the mean includes approximately 68% of the total area 1 standard deviation on both sides of the mean includes approximately 68% of the total area 2 standard deviations includes approximately 95% 2 standard deviations includes approximately 95% 3 standard deviations includes approximately 99% 3 standard deviations includes approximately 99%

Characteristics of the Normal Curve Values on the horizontal axis are Z values ranging from 0< to <1 (probability units) Values on the horizontal axis are Z values ranging from 0< to <1 (probability units) The mean is the center and the values in Standard Deviations account for proportions of the population The mean is the center and the values in Standard Deviations account for proportions of the population 1 SD = 68% of the sample 2 SD= 95% of the sample 3 SD = 99% of the sample

Characteristics of the Normal Distribution Normal distribution is completely determined by the parameters  and  Normal distribution is completely determined by the parameters  and  Different values of  shift the distribution along the x-axis Different values of  shift the distribution along the x-axis Different values of  determine degree of flatness or peakedness of the graph Different values of  determine degree of flatness or peakedness of the graph

Applications of Normal Distribution Frequently, data are normally distributed Frequently, data are normally distributed Essential for some statistical procedures Essential for some statistical procedures If not, possible to transform to a more normal form If not, possible to transform to a more normal form Approximations for other distributions Approximations for other distributions Because of the frequent occurrence of the normal distribution in nature, much statistical theory has been developed for it. Because of the frequent occurrence of the normal distribution in nature, much statistical theory has been developed for it.

What’s so Great about the Normal Distribution ? If you know two things, you know everything about the distribution If you know two things, you know everything about the distribution Mean Mean Standard deviation Standard deviation You know the probability of any value arising You know the probability of any value arising

Standardised Scores My diastolic blood pressure is 100 My diastolic blood pressure is 100 So what ? So what ? Normal is 90 (for my age and sex) Normal is 90 (for my age and sex) Mine is high Mine is high But how much high? But how much high? Express it in standardised scores Express it in standardised scores How many SDs above the mean is that? How many SDs above the mean is that?

Mean = 90, SD = 4 (my age and sex) Mean = 90, SD = 4 (my age and sex) This is a standardised score, or z-score This is a standardised score, or z-score Can consult tables (or computer) Can consult tables (or computer) See how often this high (or higher) score occur See how often this high (or higher) score occur 99.38% of people have lower scores 99.38% of people have lower scores

A Z-score Table

Standard Normal Distribution  Normal distribution is really family of curves determined by   and   Standard normal distribution is one with a  = 0 and  = 1  Standard normal density given by:  for -  < x <   where z = (x -  ) / 

Standard Normal Distribution To find probability that z takes on a value between any two points on the z-axis, need to find area bounded by perpendiculars erected at these points, the curve, and the z-axis To find probability that z takes on a value between any two points on the z-axis, need to find area bounded by perpendiculars erected at these points, the curve, and the z-axis Values are tabled. Values are tabled. Standard normal distribution is symmetric Standard normal distribution is symmetric

Examples of Standard Normal Distribution Height and weight Height and weight Calculate z-statistics Calculate z-statistics Pr(X < x) Pr(X < x) Pr(X > x) Pr(X > x) Pr(x 1 < X < x 2 ) Pr(x 1 < X < x 2 ) Why? Why? Determine percentiles Determine percentiles Comparisons between different distributions Comparisons between different distributions

Normal Distributions Go Wrong Wrong shape Wrong shape Non-symmetrical Non-symmetrical Skew Skew Too fat or too narrow Too fat or too narrow Kurtosis Kurtosis Aberrant values Aberrant values Outliers Outliers

Effects of Non-Normality Skew Skew Bias parameter estimates Bias parameter estimates E.g. mean E.g. mean Kurtosis Kurtosis Doesn’t effect parameter estimates Doesn’t effect parameter estimates Does effect standard errors Does effect standard errors Outliers Outliers Depends Depends

Distributions Bell-Shaped (also known as symmetric” or “normal”) Bell-Shaped (also known as symmetric” or “normal”) Skewed: Skewed: positively (skewed to the right) – it tails off toward larger values positively (skewed to the right) – it tails off toward larger values negatively (skewed to the left) – it tails off toward smaller values negatively (skewed to the left) – it tails off toward smaller values

Kurtosis

Outliers

Dealing with Outliers Error Error Data entry error Data entry error Correct it Correct it Real value Real value Difficult Difficult Delete it Delete it

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