Measures Of Dispersion and Normal Distribution PHD Kub.pptx

introduction to measures of dispersion and measures of central tendency. a detailed explanation about normal distribution

1. Measures of Dispersion and Normal Distribution 1

2. Measures Of Dispersion and Normal Distribution By Kubra Fatima Final Year, B.D.S GDC,VJA 2

3. Contents: Introduction Normal distribution Standard Normal distribution Z-Value Dispersion Measures of dispersion Properties of measures of dispersion Range Mean deviation Standard deviation Significance of measures of dispersion in public health dentistry Conclusion 3

4. introduction When the spread of data around the central item is high, the mean or median is less significant; low spread enhances the meaningfulness of the median or mean. Thus to increase our understanding of the pattern of the data, we must also measure its dispersion . Measures of Dispersion and Normal Distribution 4

5. Normal Distribution When data is collected from a very large number of people and a frequency distribution is made with a narrow class intervals, the resulting curve is smooth and symmetrical and it is called a NORMAL CURVE or NORMAL DISTRIBUTION. It is also known as Gaussian Distribution. The normal distribution is described by the  mean  (μ) and the  standard deviation  (σ). Measures of Dispersion and Normal Distribution 5

6. The standard normal curve is ‘bell’ shaped. The curve is perfectly symmetrical based on an infinitely large number of observations. The total area of the curve is one, its mean is zero and standard deviation one. The mean, median and mode coincide. If mean >2 standard deviation, it indicates that values are normally distributed. End/tails never touch the line. Measures of Dispersion and Normal Distribution 6

7. The area under the curve of the normal distribution represents probabilities for the data. The area under the whole curve is equal to 1, or 100% Here is a graph of a normal distribution with probabilities between standard deviations (σ): Measures of Dispersion and Normal Distribution 7

8. Roughly 68.3% of the data is within 1 standard deviation of the average (from μ-1σ to μ+1σ) Roughly 95.5% of the data is within 2 standard deviations of the average (from μ-2σ to μ+2σ) Roughly 99.7% of the data is within 3 standard deviations of the average (from μ-3σ to μ+3σ) Note:  Probabilities of the normal distribution can only be calculated for intervals (between two values). Measures of Dispersion and Normal Distribution 8

9. Example: Height of Males The distribution of the height of males in the U.S. is roughly normally distributed with a mean of 70 inches and a standard deviation of 3 inches. A histogram of the height of all U.S. male reveals a bell shape:           Measures of Dispersion and Normal Distribution 9

10. Measures of Dispersion and Normal Distribution 10 An Example of Normally Distributed Data Here is a histogram of the age of Nobel Prize winners when they won the prize: The normal distribution drawn on top of the histogram is based on the population mean (μ) and standard deviation (σ) . We can see that the histogram close to a normal distribution.

11. Standard Normal Distribution also called the 'Z-distribution' and the values are called 'Z-values' (or Z-scores). Normally distributed data can be transformed into a standard normal distribution. Standardizing normally distributed data makes it easier to compare different sets of data. The standard normal distribution is used for: Calculating confidence intervals Hypothesis tests Here is a graph of the standard normal distribution with probability values (p-values) between the standard deviations:      Standardizing makes it easier to calculate probabilities. Measures of Dispersion and Normal Distribution 11

12. Z-Values Z-values express how many standard deviations from the mean a value is. The formula for calculating a Z-value is: Z=(x−μ)/σ x is the value we are standardizing,  μ is the mean , and  σ is the standard deviation . For example, if we know that: The mean height of people in Germany is 170 cm (μ) The standard deviation of the height of people in Germany is 10 cm (σ) Bob is 200 cm tall (x) Bob is 30 cm taller than the average person in Germany. 30 cm is 3 times 10 cm. So , Bob's height is 3 standard deviations larger than mean height in Germany. Using the formula: Z=x− μσ =(200−170)/10 The Z-value of Bob's height (200 cm) is 3. Measures of Dispersion and Normal Distribution 12

13. Interpretation of Z-value : A z value of 0 means the data point is equal to the mean. A positive z value the data point is above the mean. A negative z value means the data point is below the mean. The larger the absolute value of the z value the farther away the data point is from the mean. Measures of Dispersion and Normal Distribution 13

14. Different Mean and Standard Deviations The mean describes where the center of the normal distribution is. Here is a graph showing three different normal distributions with the  same  standard deviation but different means. The standard deviation describes how spread out the normal distribution is. Here is a graph showing three different normal distributions with the  same  mean but different standard deviations. The purple curve has the biggest standard deviation and the black curve has the smallest standard deviation. The area under each of the curves is still 1, or 100%. Measures of Dispersion and Normal Distribution 14

15. Dispersion Dispersion is the degree of spread or variation of the variable about a central value. Measures of Dispersion and Normal Distribution 15

16. Measures of dispersion Measures of dispersion helps to know how widely the observations are spread on the side of the average. The most common measures of dispersion used in dental science are: 1. Range 2. Mean Deviation 3. Standard Deviation Measures of Dispersion and Normal Distribution 16

17. Significance and properties of Measures of Dispersion A good measure of dispersion should possess the following properties: It should be simple to understand It should be easy to compute It should be rigidly defined It should be based on each and every item of the distribution It should be amenable to further algebraic treatment It should have sampling stability Extreme items should not unduly affect it Measures of Dispersion and Normal Distribution 17

18. Range It is defined as the difference between the value of the largest item and the value of the smallest item. Measures of Dispersion and Normal Distribution 18

19. Gives no information about the values that lie between the extreme values. Its not based on all the items and is subject to fluctuations of considerable magnitude from samples to sample. It is badly affected by the extreme values Measures of Dispersion and Normal Distribution 19 Disadvantages Advantages Quick and easy to calculate A reasonably good indicator

20. Mean Deviation It is the average of the deviation from the arithmetic mean. It is given by, M.D.= Wh ere - sum of X - arithmetic mean Xi - value of each observation in the data n - number of observations in the data   Measures of Dispersion and Normal Distribution 20

21. Example: For instance, consider a dataset: 10, 15, 20, 25. The mean is (10+15+20+25)/4 = 17.5. Deviations are (-7.5), (-2.5), 2.5, and 7.5, respectively. Absolute deviations are 7.5, 2.5, 2.5, and 7.5. The sum of these absolute deviations is 20 divided by 4 (total observations) equals a mean deviation of 5. Measures of Dispersion and Normal Distribution 21

22. Advantages Takes into account all values in the sample Easy to understand Simple to compute Less affected by extreme values It has fixed value Disadvantages The absolute value signs are very cumbersome in mathematical equations Measures of Dispersion and Normal Distribution 22

23. Standard Deviation It is the most important and widely used measure of studying dispersion. It is also known as root mean square deviation because it is the square root of the mean of the squared deviations from the arithmetic mean. Greater the standard deviation, greater will be the magnitude of dispersion from the mean. A small standard deviation means a higher degree of uniformity of the observations. S.D. =   Measures of Dispersion and Normal Distribution 23

24. Standard Deviation Standard deviation (σ) measures how far a 'typical' observation is from the average of the data (μ). Standard deviation is important for many statistical methods. Here is a histogram of the age of all 934 Nobel Prize winners up to the year 2020, showing  standard deviations :      Measures of Dispersion and Normal Distribution 24

25. How to calculate Standard Deviation? Calculate the mean of the series. Take the deviations of the items from the mean. Square the deviations and add them up. Divide the result by the total number of observations Obtain the square root. This gives the standard deviation. Measures of Dispersion and Normal Distribution 25

26. Advantages It is the most reliable measure of dispersion. It is the most widely used measure of dispersion. Its computation is based on all the observations. Disadvantages It is relatively difficult to calculate and understand. It cannot be used for comparing the dispersion of two or more series given in different units. It is affected very much by the extreme values. Measures of Dispersion and Normal Distribution 26

27. Understanding variability Identifying inequalities Assessing risk factors Informing resource allocation Evaluating program effectiveness Enhancing surveillance Measures of Dispersion and Normal Distribution 27 Significance of measures of distribution in public health dentistry

28. Conclusion By applying measures of dispersion and normal distribution, public health dentistry professionals can gain a deeper understanding of oral health outcomes, identify areas of improvement, and develop more effective interventions to promote oral health equity. Measures of Dispersion and Normal Distribution 28

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