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Npc, skewness and kurtosis
The document discusses properties of the normal probability distribution or "normal curve". It describes: 1) The normal curve is bell-shaped and symmetrical about the mean, with the mean, median and mode falling at the same point. 2) Approximately 68% of the values fall within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations, according to the empirical rule. 3) Skewness measures asymmetry - a curve can be positively skewed if the tail extends toward higher values, negatively skewed if the tail extends toward lower values, or symmetrical. 4) Kurtosis measures peakedness - a distribution can be platykurtic
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Npc, skewness and kurtosis
- 1. NORMAL PROBABILITY CURVE K.THIYAGU, Assistant Professor,Department of Education, Central University of Kerala, Kasaragod
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- 3. The normal probabilitydistribution or the “normal curve” is often called the Gaussian distribution, Carl Gauss
- 4. ❑ The normaldistribution was first discovered by De- moivre in 1733 to solve problems in games or chances. ❑ Later it was applied in natural and social science by the French Mathematician La Place (1949). ❑ This concept further developed by Gauss. ❑ The literal meaning of the term normal is average. NPC
- 5. J.P. Guilford hasdefined normal probability curve comprehensively. “Normal probability curve is well defined, well structure, mathematical curve, having a distribution of the scores with mean, median and modes are equal. NPC
- 6. Bell Shaped Curve Theshape of the curve is like that of a bell.
- 7. Mean = Median= Mode Mean, median and mode carry the equal value. / same numerical value mean, median and mode Therefore they fall at the same point on the curve
- 8. Unimodal Since themean , median and mode lye at one point of the curve it is unimodal in nature.
- 9. Perfectly Symmetricality Itmeans the curve inclines towards both sides equally from the centre of the curve. Thus we get equal halves on both sides from the central point. The curve is not skewed. Therefore the values of the measure of skewness is zero
- 10. A normal curveis symmetric about the mean. Each of the two shaded areas is .5 or 50% .5.5 μ x
- 11. Asymptotic The curvedoes not touch the base or OX axis on both sides. Thus it extends from negative infinity to positive infinitive.
- 12. Distance of theCurve For practical purpose the base line of the curve is divided into six sigma distance from Most of the cases I.e.99.73% are covered within such distance
- 13. Maximum Ordinate Maximumordinate of the curve occurs at the mean. I.e. where Z1=0 and the value of the highest ordinate is 0.3989. The height of the ordinate at 1sigma is 0.2420 2 sigma = 0.0540 3 sigma = 0.0044
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- 15. 68-95-99.7 Rule • Forany normal curve with mean mu and standard deviation sigma: • 68 percent of the observations fall within one standard deviation sigma of the mean. • 95 percent of observation fall within 2 standard deviations. • 99.7 percent of observations fall within 3 standard deviations of the mean.
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- 17. x - sx x + s 68% within 1 standard deviation 34% 34% The Empirical Rule
- 18. x - 2sx - s x x + 2sx + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations 13.5% 13.5% The Empirical Rule
- 19. x - 3sx - 2s x - s x x + 2s x + 3sx + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean 0.1% 0.1% 2.4% 2.4% 13.5% 13.5% The Empirical Rule
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- 21. Properties (cont.) • Hasa mean = 0 and standard deviation = 1. • General relationships: ±1 s = about 68.26% ±2 s = about 95.44% ±3 s = about 99.72% -5 -4 -3 -2 -1 0 1 2 3 4 5 68.26% 95.44% 99.72%
- 22. Analysis of Scalevalue The scale values are analysed as Z1=(X-M)/sigma Where z1 = 0 and range of mean plus or minus Z is equal to mean plus or minus sigma
- 23. Points of infection The points of infection are each plus or minus one sigma from above and below the mean. The curve changes from convex to concave at these points with the baseline.
- 24. Various Measures In thenormal curve Quartile deviation Q = Proper error = 0.6745a Mean deviation, AD = 0.7979a Skewness = 0 Kurtosis = 0.263
- 25. APPLICATIONS OF THENORMAL PROBABILITY CURVE To normalize a frequency distribution. It is an important step in standardizing a psychological test or inventory. To test the significance of observations in experiments, findings them relationships with the chance fluctuations or errors that are result of sampling procedures. To generalize about population from which the samples are drawn by calculating the standard error of mean and other statistics. To compare two distributions. The NPC is used to compare two distributions. To determine the difficulty values. The Z scores are used to determine the difficulty values of test items. To classify the groups. The Normal Probability Curve (NPC) is used for classifying the groups and assigning grades to individuals. To determine the level of significance. The levels of significance of statistics results are determined in terms of NPC limits. To scale responses to opinionnaires, judgement, ratings or rankings by transforming them numerical values.
- 26. Skewness K.THIYAGU, Assistant Professor,Department of Education, Central University of Kerala, Kasaragod
- 27. Skewness ❑ Skewness meansasymmetrical nature / lack of symmetry ❑ The degree of departure from symmetry is called skewness. ❑ The distribution in which mean, Median and Mode fall on different points or different places is known as skewed distribution and this tendency of distribution is known as skewness.
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- 30. POSITIVELY SKEWED CURVE/ POSITIVE SKEWNESS When the curve inclines more towards right we ascertain the positive skewness. If the longer tail of the distribution is towards the higher values or upper sides, the skewness is positive.
- 31. When the curveinclines more towards right Positive or Right Skew Distribution
- 33. Properties of Positively Skewed M>Mdn>Mo Low Achievers or failures, Subject is difficult Teaching is ineffective Students did not prepare well for the examination Strict Valuation Question paper consists of questions having higher difficulty level,
- 34. NEGATIVELY SKEWED CURVE Whenthe curve inclines more to the left skewness becomes negative. If the longer tail of the distribution is towards the lower values or lower sides, the skewness is negative.
- 35. When the curveinclines more to the left skewness becomes negative Negative or Left Skew Distribution
- 36. Properties of Negatively SkewedCurve M < Mdn < Mo Intelligent and studious students Subject is easy Hard preparation Teaching is very effective Examiner is very liberal in giving marks Easy question paper
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- 38. Skewness Skewness = -.5786 Suggestingslight left skewness. Skewness = 1.944 Suggesting strong right skewness.
- 39. C.I. NORMAL CURVE POSITIVELY SKEWED NEGATIVELY SKEWED ff f 91 - 100 2 2 20 81 – 90 3 2 10 71 - 80 10 3 10 61 – 70 20 3 3 51 – 60 10 10 3 41 – 50 3 10 2 31 - 40 2 20 2 Example
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- 42. Kurtosis The Kurtosisof a distribution refers to its ‘curvedness’ or ‘peakedness’. The distributions may have the same mean and the same variance and may be equally skewed, but one of them may be more peaked than the other. Kurtosis refers to peakness or flatness of a normal curve.
- 43. Kurtosis In somedistribution the values of mean, median and mode are the same. But if a curve is drawn from the distribution then the height of curve is either more or less than the normal probability curve, since such type of deviation is related with the crest of the curve, it is called kurtosis. The curves with kurtosis are of two types 1) Leptokurtic curve and 2) Playtykurtic curve
- 44. Kurtosis Leptokurtic: highand thin Mesokurtic: normal in shape Platykurtic: flat and spread out Leptokurtic Mesokurtic Platykurtic Peak Kurtosis Highest Peak Leptokurtic Medium Peak Mesokurtic Smallest or Flattest Peak Platykurtic
- 46. Leptokurtic ❑ Lepto meansslender or narrow. ❑ When the curve is more peaked than normal one it is called leptokurtic curve. ❑ if maximum frequencies in a distribution are concentrated around the mean, then the number of frequencies falling between -1 to +1 is more than the frequencies falling within the range in case of normal probability curve ❑ If Kurtosis is less than 0.263 the distribution is leptokurtic.
- 47. Inferences from theleptokurtic curve Homogenous group students (Most Average students) Intelligent and dull students are less in number. The examiner has allotted average marks to most of the students The teacher may have used only one method of teaching for the whole class by neglecting individual difference Questions with more or less similar difficulty level may have been included in the question paper by the examiner.
- 48. Platykurtic Platy meansflat, broad or wide. When the curve is more flattered the distribution will be called as ‘platykurtic’. In exactly opposite condition the scores are not concentrated around the mean. Therefore, the number of frequencies falling between -1 to +1 is lower in comparison with the normal probability curve and the curve drawn for this distribution is known as platykurtic curve. If Kurtosis is greater than 0.263 the distribution is platykurtic.
- 49. Inferences from theplaytykurtic curve The teaching is ineffective. Valuation of papers may have been improper. Discrimination index of questions asked in exam may be of higher order. The group of students is heterogeneous. There may have been variation in the difficulty level of question.
- 50. REASONS FOR DIVERGENCEFROM NORMALITY / FACTORS AFFECTIVING DIVEREGENCE IN THE NORMAL CURVE Improper selection of sample / Biased selection of sample: Improper construction of a test / Test construction Error: Improper administration of a test / Test Administration Error: Using Unsuitable test:
- 51. kurtosis: the proportionof a curve located in the center, shoulders and tails How fat or thin the tails are leptokurtic no shoulders platykurtic wide shoulders
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