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Understanding Mean, Median, Percentiles, and Quartiles in Data Analysis

This article, prepared by Mrs. Salwa Kamel, provides an in-depth exploration of essential statistical concepts including mean, median, percentiles, quartiles, and measures of variation. It explains how to find the 60th percentile, determine quartiles using ordered data, and calculate the interquartile range while addressing outliers. The article also covers the normal distribution and skewness, offering practical examples to illustrate these concepts. Key formulas for standard deviation and variance are included, helping readers to accurately analyze data distributions.

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Understanding Mean, Median, Percentiles, and Quartiles in Data Analysis

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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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