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Lesson 8 zscore

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  • 1. Introduction to Statistics for Built Environment Course Code: AED 1222 Compiled by DEPARTMENT OF ARCHITECTURE AND ENVIRONMENTAL DESIGN (AED) CENTRE FOR FOUNDATION STUDIES (CFS) INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIA
  • 2. Lecture 10 Normal Curve Today’s Lecture:  Calculating the z-score  Interpreting the z-score  Areas under the normal curve  Skewness
  • 3. Z-Scores – a Measure of Position • The z -score is also called the standard score. • It is an ordinary score converted so that it better describes that score's place in its distribution. • It denotes the number of standard deviations a data value is above the mean (if +) or below the mean (if -). • Why z-score? – As a scale. – Provides a generalized standard of comparison.
  • 4. Calculating the z-score Z-score formula for sample data Z-score formula for population data Z = s x - x Z = x - µ s This value is a measure of the distance in standard deviations of a data value from the mean.
  • 5. 5 Calculating Z – Scores: an example Mansur is interested to become a salesperson to gain basic marketing skills. During an interview, he took a sales skill test and scored 135. Does this indicate he has the potential for sales? If average score of this test is 100, his score (135) is above average. But how much is above average? Slightly, moderately or dramatically? How high is Mansur’s score as compared to anyone else who took the same test?
  • 6. Given standard deviation is 25, convert Mansur’s score to a z- score. Z = 1.4 Original score New score/Z-Score Z = x - µ s Z = 135 - 100 25 Calculating z-scores cont.
  • 7. 7 The same case, continued At the interview, Mansur noticed there’s another vacant position in research analysis. He thought why not having a try in case he does not make it in the first interview. In a test to apply for this post, he scored 85. The average score (mean) of this test is 60 and the standard deviation is 10. Calculating z-scores cont.
  • 8. Let’s see Mansur’s z-score in the second test>> Z = x - µ s Z = 85 - 60 10 Z = 2.5 Original score New score/Z-Score So Mansur scored 2.5 std. deviations above the mean in research analysis test but he scored only 1.4 std. deviations above the mean in sales skill test. It seems that Mansur’s inclinations is more towards research analysis. Calculating z-scores cont.
  • 9. Interpreting z-scores z-score < 0 Is when a data value is less than the mean z -score > 0 Is when a data value is greater than the mean z -score = 0 Is when a data value is equal to the mean
  • 10. A normal distribution is a bell-shaped distribution of data where the mean, median and mode all coincide. Chebyshev’s theorem applies to any distribution regardless of its shape. However, when a distribution is bell- shaped, the following statements which make up the empirical rule, are true.
  • 11. Approximately: •68% of the data values will fall within 1 sd. Of the mean. •95% of the data values will fall within 2 sd. Of the mean. •99.7% of the data values will fall within 3 sd. Of the mean.
  • 12. For example: The score on a national achievement exam have a mean of 480 and a deviation of 90. If these scores are normally distributed, then approximately 68% will fall between 390 and 570 (480+90=570 and 480-90=390) 480390 570
  • 13. Skewness In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Mean Mode
  • 14. Positively skewed/ skewed to the right If there are extreme values towards the positive end of a distribution, the distribution is said to be positively skewed. In a positively skewed distribution, the mean is greater than the mode. For example:
  • 15. Negatively skewed/ skewed to the left A negatively skewed distribution, on the other hand, has a mean which is less than the mode because of the presence of extreme values at the negative end of the distribution.
  • 16. Calculating skewness There are a number of ways of measuring skewness: Pearson’s coefficient of skewness = mean – mode = 3(mean – median) Standard deviation Standard deviation The value determines whether the data distribution is positively skewed or negatively skewed. If the value is positive, it means the distribution is positively/rightly skewed. If the value is negative, it means the distribution is negatively/ leftly skewed.