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Prepared by Mrs. Salwa Kamel

Revisoin. Prepared by Mrs. Salwa Kamel. Mean – Mode- Median. Percentiles. The p th percentile in an ordered array of n values is the value in i th position, where. Example: Find the position of 60 th percentile in an ordered array ( arrangement ,) of 19 values?

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Prepared by Mrs. Salwa Kamel

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  1. Revisoin Prepared byMrs. SalwaKamel

  2. Mean – Mode- Median

  3. Percentiles • The pth percentile in an ordered array of n values is the value in ith position, where Example:Find the position of 60th percentile in an ordered array (arrangement,)of 19 values? It is the value in 12th position:

  4. Quartiles • Quartiles split the ranked data into 4 equal groups 25% 25% 25% 25% Q1 Q2 Q3 Example: Find the first quartile Sample Data in Ordered Array: 11 12 13 16 16 17 18 21 22 (n = 9) Q1 = 25th percentile, so find the (9+1) = 2.5 position so use the value half way between the 2nd and 3rd values, So Q1 = 12.5 25 100

  5. Example 5

  6. Interquartile Range • Can eliminate some outlier problems by using the interquartile range • Eliminate some high-and low-valued observations and calculate the range from the remaining values. • Interquartile range = 3rd quartile – 1st quartile

  7. Comparing Coefficient of Variation • Stock A: • Average price last year = $50 • Standard deviation = $5 • Stock B: • Average price last year = $100 • Standard deviation = $5 Both stocks have the same standard deviation, but stock B is less variable relative to its price

  8. Best Measure of Center

  9. Sample Standard Deviation Formula (x – x)2 s= n –1

  10. Population Standard Deviation (x – µ) 2  = N This formula is similar to the previous formula, but instead, the population mean and population size are used.

  11. Variance - Notation s=sample standard deviation s2= sample variance =population standard deviation 2=population variance

  12. maximum value + minimum value Midrange= 2 Midrange • the value midway between the maximum and minimum values in the original data set

  13. Example • Find Range and midrange for the following data a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10

  14. Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half.

  15. Skewed to the left (also called negatively skewed) have a longer left tail, mean and median are to the left of the mode

  16. Skewed to the right (also called positively skewed) have a longer right tail, mean and median are to the right of the mode

  17. The Relative Positions of the Mean, Median and the Mode 3-17

  18. Measure of Skewness • Describes the degree of departures of the distribution of the data from symmetry. • The degree of skewness is measured by the coefficient of skewness, denoted as SK and computed as, Remark: a) If SK > 0, then the distribution is skewed to the right. b) SK < 0, then the distribution of the data set is skewed to left. c) If SK = 0, then the distribution is symmetric. a symmetric distribution has SK=0 since its mean is equal to its median and its mode.

  19. Example: Consider again the out – of – state tuition rates for the six school sample from Pennsylvania. 4.9 6.3 7.7 8.9 7.7 10.3 11.7 1) Determine the following: 1. Range 2. Inter – quartile Range 3. Standard Deviation 4. Variance 2) Determine the direction of skewness of the preceding data.

  20. Example 3-20

  21. x – µ x – x z = s z =  Measures of Position z Score Sample Population Round z scores to 2 decimal places

  22. Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score < –2 or z score > 2

  23. Standard Normal Scores How many standard deviations away from the mean are you? Standard Score (Z) = “Z” is normal with mean 0 and standard deviation of 1. Observation – meanStandard deviation Z Score It is a standard score that indicates how many SDs from the mean a particular values lies. Z = Score of value – mean of scores divided by standard deviation. 23

  24. Standard Normal Scores Example: Male Blood Pressure,mean = 125, s = 14 mmHg 1) BP = 167 mmHg (Observation) 2) BP = 97 mmHg Note that: Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score < –2 or z score > 2 24

  25. Standardizing Data: Z-Scores Ordinary values: –2 ≤ z score ≤ 2 Unusual Values: z score < –2 or z score > 2 25

  26. Five Number Summary Median Q1 Smallest Q3 Largest 26

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