Discuss divergence in normality with the help of suitable diagram and describe the factors causing divergence in the normal distribution. Discuss how divergence in normality is measured

Generally two types of divergence occur in the normal curve:

(i) Skewness: A distribution is said to be ‘skewed’ when the mean and median fall at different points in thedistribution and the balance i.e. the point of center of gravity is shifted to one side or the other to left or right. In anormal distribution the mean equals, the median exactly and the skewness is of course zero (SK = o).

There are two types of skewness which appear in the normal curve:

(a) Negative Skewiiess: Distribution said to be skewed negatively or to the left when scores are massed at thehigh end of the scale, i.e. the right side of the curve are spread out more gradually toward the low end i.e. the leftside of the curve. In negatively skewed distribution the value of median will be higher than that of the value of themean.

(b) Positive Skewness: Distribution are skewed positively or to the ri adually toward the high or right end.

(2) Kurtosis: The term kurtosis refers to (the divergence) in the height of the curve, especially in the peakness.There are two types of divergence in the peakness of the curve

(a) Leptokurtosis: Suppose you have a normal curve which is made up of a steel wire. If you push both theends of the wire curve together. What would happen in the shape of the curve? Probably your answer may be that bypressing both the ends of the wire curve, the curve become more peeked i.e. its top become narrower than the normal curve and scatterdness in the scores or area of the curve shrink towards the center. Thus, in a Leptokurticdistribution, the frequency distribution curve is more peaked than to the normal distribution curve.

(b) Playkurtosis: Now suppose we put a heavy pressure on the top of the wire made normal curve. What wouldbe the change in the shape of the curve? Probabliy you may asy that the top of the curve become flatter than to thenormal.

Thus, a distribution of flatter peak than to the normal is known platykurtosis distribution. When the distributionand related curve is normal, the vain of kurtosis is 0.263 (KU = 0.263).If the value of the KU is greater than 0.263,the distribution and related curve obtained will be platkurtic. When the value of KU is less than 0.263, the distributgionand related curve obtained will be Leptokurtic.

Factors causing divergence in the normal distribution/normal curve: The factors causing divergence in thenormal distribution/normal curve are as follows:

(1) Selection of the Sample: Selection of the subjects (individuals) produces skewness and kurtosis in thedistribution. If the sample size is small or sample is biased one, skewness is possible in the distribution of scoresobtained on the basisof selected sample or group individuals.

If the scores made by small and homogeneous groups are likely to yield narrow and leptokurtic distribution.Scores from small and highly heterogeneous groups yield platykurtic distribution.

(2) Unsuitable on Poorly Made Tests: If the measuring tool or test is inappropriate or poorly made, theasymmetry is possible in the distribution of scores. If a test is too easy, scores will pile up at the high end of thescale, whereas the test is too hard, scores will pile up at the low end of the scale.

(3) The Trait being measured is Non-Normal: Skewness or Kurtosis or both will appear when there is a reallack of normality in the trait being meassured, e.g. interest, attitude, suggestibility, deaths in old age or early-childhood due to certain degenerative disceases etc.

(4) Errors in the Construction and Administration of Tests: The unstandardised with poor item-analysis testmay cause asymmetry in the distribution of the scores. Similarly, while administrating the test, the unclear instructions,

i.e. errors in timings, errors in the scoring practice and motivation to complete the test-all these factors may causeskewness in the distribution.

Measurement of divergence in normality: In psychology and education, the divergence in normal distributionnormal curve has a significant role in construction of the ability and mental tests and to test the representativenessof a sample taken from a large population. Further, the divergence in the distribution of scores or measurementsobtained of a certain population reflects some important information about the trait of population measured. Thus,there is a need to measure the two divergence, i.e. skewness and kurtosis of the distribution of the scores.

Measuring Skewness: There are two methods to study the skewness in a distribution. These are as follows:

(1) Observation Method: There is a simple method of detecting the directions of skewness by the inspectionof frequency polygon prepared on the basis of the scores obtained regarding a trait of the population or a sampledrawn from a population.

Looking al the tails of the frequency polygon of the distribution obtained if longer tail of the curve is towardsthe higher value or upper side or right side to the centre or mean, the kewness is positive. If the longer tail is towardsthe lower values or lower side or lefUtffie mean, the skewness is necative.

(2) Statistical Method: To know the skewness in the distribtution, we may also use the statistical method. Forthe purpose, we use measures of central tendency specifically mean and median values and use the followingformula.

3Mean–Median/σ

Another measure of skewness based on precntile values, is an under:

Sk = (P30 – P10)/2 – P50

Here, it is to be kapt in mind that the above two measures are not mathematically equivalent. A normal curve hasthe value of Sk = 0. Deviations from normality can be negative and positively direction leading to negatively skewedand positively skewed distributions respectively.

Measuring Kurtosis: For juicing whether a distribution lacks normal symmetry or peakedness; it may detectedmay detected by inspection of the frequency polygon obtained. If a peak of curve is thin and sides are narrow to thecentre, the distribution is leptokurtic and if the peak of the frequency distribution is too flat and sides of the curveare deviating from the centre towards ± 4 or ± 5 then the distribution is platykurtic.

Kurtosis can be measured by following formula us ing percentile values.

A normal distribution has KU = 0.263. If the value of KU is less than 0.263 (KU < 0.263), the distribution isleptokurtic and if KU is greater than 0.263 (KU > 0.263), the distribution is platykurtic.

Generally two types of divergence occur in the normal curve:

(i) Skewness:A distribution is said to be ‘skewed’ when the mean and median fall at different points in the distribution and the balance i.e. the point of center of gravity is shifted to one side or the other to left or right. In a normal distribution the mean equals, the median exactly and the skewness is of course zero (SK = o).There are two types of skewness which appear in the normal curve:

(a) Negative Skewiiess:Distribution said to be skewed negatively or to the left when scores are massed at the high end of the scale, i.e. the right side of the curve are spread out more gradually toward the low end i.e. the left side of the curve. In negatively skewed distribution the value of median will be higher than that of the value of the mean.(b) Positive Skewness:Distribution are skewed positively or to the ri adually toward the high or right end.(2) Kurtosis:The term kurtosis refers to (the divergence) in the height of the curve, especially in the peakness. There are two types of divergence in the peakness of the curve(a) Leptokurtosis:Suppose you have a normal curve which is made up of a steel wire. If you push both the ends of the wire curve together. What would happen in the shape of the curve? Probably your answer may be that by pressing both the ends of the wire curve, the curve become more peeked i.e. its top become narrower than the normal curve and scatterdness in the scores or area of the curve shrink towards the center. Thus, in a Leptokurtic distribution, the frequency distribution curve is more peaked than to the normal distribution curve.(b) Playkurtosis:Now suppose we put a heavy pressure on the top of the wire made normal curve. What would be the change in the shape of the curve? Probabliy you may asy that the top of the curve become flatter than to the normal.Thus, a distribution of flatter peak than to the normal is known platykurtosis distribution. When the distribution and related curve is normal, the vain of kurtosis is 0.263 (KU = 0.263).If the value of the KU is greater than 0.263, the distribution and related curve obtained will be platkurtic. When the value of KU is less than 0.263, the distributgion and related curve obtained will be Leptokurtic.

Factors causing divergence in the normal distribution/normal curve:The factors causing divergence in the normal distribution/normal curve are as follows:(1) Selection of the Sample:Selection of the subjects (individuals) produces skewness and kurtosis in the distribution. If the sample size is small or sample is biased one, skewness is possible in the distribution of scores obtained on the basisof selected sample or group individuals.If the scores made by small and homogeneous groups are likely to yield narrow and leptokurtic distribution. Scores from small and highly heterogeneous groups yield platykurtic distribution.

(2) Unsuitable on Poorly Made Tests:If the measuring tool or test is inappropriate or poorly made, the asymmetry is possible in the distribution of scores. If a test is too easy, scores will pile up at the high end of the scale, whereas the test is too hard, scores will pile up at the low end of the scale.(3) The Trait being measured is Non-Normal:Skewness or Kurtosis or both will appear when there is a real lack of normality in the trait being meassured, e.g. interest, attitude, suggestibility, deaths in old age or early-childhood due to certain degenerative disceases etc.(4) Errors in the Construction and Administration of Tests:The unstandardised with poor item-analysis test may cause asymmetry in the distribution of the scores. Similarly, while administrating the test, the unclear instructions,i.e. errors in timings, errors in the scoring practice and motivation to complete the test-all these factors may cause skewness in the distribution.

Measurement of divergence in normality: In psychology and education, the divergence in normal distribution normal curve has a significant role in construction of the ability and mental tests and to test the representativeness of a sample taken from a large population. Further, the divergence in the distribution of scores or measurements obtained of a certain population reflects some important information about the trait of population measured. Thus, there is a need to measure the two divergence, i.e. skewness and kurtosis of the distribution of the scores.

Measuring Skewness:There are two methods to study the skewness in a distribution. These are as follows:(1) Observation Method:There is a simple method of detecting the directions of skewness by the inspection of frequency polygon prepared on the basis of the scores obtained regarding a trait of the population or a sample drawn from a population.Looking al the tails of the frequency polygon of the distribution obtained if longer tail of the curve is towards the higher value or upper side or right side to the centre or mean, the kewness is positive. If the longer tail is towards the lower values or lower side or lefUtffie mean, the skewness is necative.

(2) Statistical Method:To know the skewness in the distribtution, we may also use the statistical method. For the purpose, we use measures of central tendency specifically mean and median values and use the following formula.3 Mean – Median/σ

Another measure of skewness based on precntile values, is an under:

Sk = (P30 – P10)/2 – P50

Here, it is to be kapt in mind that the above two measures are not mathematically equivalent. A normal curve has the value of Sk = 0. Deviations from normality can be negative and positively direction leading to negatively skewed and positively skewed distributions respectively.

Measuring Kurtosis:For juicing whether a distribution lacks normal symmetry or peakedness; it may detected may detected by inspection of the frequency polygon obtained. If a peak of curve is thin and sides are narrow to the centre, the distribution is leptokurtic and if the peak of the frequency distribution is too flat and sides of the curve are deviating from the centre towards ± 4 or ± 5 then the distribution is platykurtic.Kurtosis can be measured by following formula us ing percentile values.

Ku = Q/P – P

Where Q = quartile deviation, i.e.

P10 = 10th percentile

= 90th percentile

P90 = 90th percentile

A normal distribution has KU = 0.263. If the value of KU is less than 0.263 (KU < 0.263), the distribution is leptokurtic and if KU is greater than 0.263 (KU > 0.263), the distribution is platykurtic.