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Normal distribution

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Sylvestre Gahungu

This document discusses the normal distribution and how to standardize data. It explains that normally distributed data forms a bell curve around a central mean. It also describes how the standard deviation measures how spread out data is, with 68% of values within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. The document demonstrates how to convert a value into a z-score or standard score by subtracting the mean and dividing by the standard deviation, effectively standardizing the data.

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EXPERIMENTAL PROBABILITY NORMAL DISTRIBUTION
NORMAL DISTRIBUTION There are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: The "Bell Curve" is a Normal Distribution. And the yellow histogram shows some data that follows it closely, but not perfectly (which is usual).
DATA DISTRIBUTION Data can be "distributed" (spread out) in different ways It can be spread out more on the left Or more on the right Or it can be all jumbled up
NORMAL DISTRIBUTION (CONT¡¯D) In real life, many things closely follow a Normal Distribution: ? heights of people ? size of things produced by machines ? errors in measurements ? blood pressure ? marks on a test We say that the data is normally distributed
STANDARD DEVIATION ? The Standard Deviation is a measure of how spread out numbers are (read that page for details on how to calculate it). ? When we calculate the standard deviation we find that generally: ? 68% of values are within 1 standard deviation of the mean ? 95% of values are within 2 standard deviations of the mean ? 99.7% of values are within 3 standard deviations of the mean
Example: 95% of students at school are between 1.1m and 1.7m tall. Answer: ? Assuming this data is normally distributed we can calculate the mean and standard deviation The mean is halfway between 1.1m and 1.7m: Mean = (1.1m + 1.7m) / 2 = 1.4m ? 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so: 1 standard deviation = (1.7m-1.1m) / 4 = 0.6m / 4 = 0.15m
STANDARDISING A NORMAL DISTRIBUTION Standard scores Example: In that same school one of your friends is 1.85m tall You can see on the bell curve that 1.85m is 3 standard deviations from the mean of 1.4, so: Your friend's height has a "z-score" of 3.0 It is also possible to calculate how many standard deviations 1.85 is from the mean: 1.85 cm is 0.45 cm (1.85 ¨C 1.4) far from the mean; 3 standard deviations (0.45 ¡Â 0.015) from the mean
STANDARDISING A NORMAL DISTRIBUTION So to convert a value to a Standard Score ("z-score"): ? first subtract the mean, ? then divide by the Standard Deviation And doing that is called "Standardizing": We can take any Normal Distribution and convert it to The Standard Normal Distribution.
STANDARDISING A NORMAL DISTRIBUTION (CONT¡¯D) Example: Travel Time A survey of daily travel time had these results (in minutes):A survey of daily travel time had these results (in minutes): 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34 The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes (you can copy and paste the values into the Standard Deviation Calculator if you want). Convert the values to z-scores ("standard scores"). To convert 26: ? = ?????.? ??.? = ?1.12; So 26 is ?1.12 Standard Deviations from the Mean Original Value Calculation Standard Score (z-score) 26 ?? ? ??. ? ??. ? = ?1.12 33 ?? ? ??. ? ??. ? = ?0.51 65 ?? ? ??. ? ??. ? = +2.30
STANDARDISING A NORMAL DISTRIBUTION (CONT¡¯D) And here they are graphically: The rest of the scores can be calculated using the same formula. (You may try calculating those)
STANDARDISING FORMULA The z-score formula that we have been using is: z is the "z-score" (Standard Score) x is the value to be standardized ¦Ì ('mu") is the mean ¦Ò ("sigma") is the standard deviation
IN MORE DETAILS Here is the Standard Normal Distribution with percentages for every half of a standard deviation, and cumulative percentages:
STANDARD NORMAL DISTRIBUTION TABLE Open the link below to access the table: https://www.mathsisfun.com/data/standard-normal-distribution- table.html Exercises Answer the questions at the bottom of this page: https://www.mathsisfun.com/data/standard-normal- distribution-table.html

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Normal distribution

  • 2. NORMAL DISTRIBUTION There are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: The "Bell Curve" is a Normal Distribution. And the yellow histogram shows some data that follows it closely, but not perfectly (which is usual).
  • 3. DATA DISTRIBUTION Data can be "distributed" (spread out) in different ways It can be spread out more on the left Or more on the right Or it can be all jumbled up
  • 4. NORMAL DISTRIBUTION (CONT¡¯D) In real life, many things closely follow a Normal Distribution: ? heights of people ? size of things produced by machines ? errors in measurements ? blood pressure ? marks on a test We say that the data is normally distributed
  • 5. STANDARD DEVIATION ? The Standard Deviation is a measure of how spread out numbers are (read that page for details on how to calculate it). ? When we calculate the standard deviation we find that generally: ? 68% of values are within 1 standard deviation of the mean ? 95% of values are within 2 standard deviations of the mean ? 99.7% of values are within 3 standard deviations of the mean
  • 6. Example: 95% of students at school are between 1.1m and 1.7m tall. Answer: ? Assuming this data is normally distributed we can calculate the mean and standard deviation The mean is halfway between 1.1m and 1.7m: Mean = (1.1m + 1.7m) / 2 = 1.4m ? 95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so: 1 standard deviation = (1.7m-1.1m) / 4 = 0.6m / 4 = 0.15m
  • 7. STANDARDISING A NORMAL DISTRIBUTION Standard scores Example: In that same school one of your friends is 1.85m tall You can see on the bell curve that 1.85m is 3 standard deviations from the mean of 1.4, so: Your friend's height has a "z-score" of 3.0 It is also possible to calculate how many standard deviations 1.85 is from the mean: 1.85 cm is 0.45 cm (1.85 ¨C 1.4) far from the mean; 3 standard deviations (0.45 ¡Â 0.015) from the mean
  • 8. STANDARDISING A NORMAL DISTRIBUTION So to convert a value to a Standard Score ("z-score"): ? first subtract the mean, ? then divide by the Standard Deviation And doing that is called "Standardizing": We can take any Normal Distribution and convert it to The Standard Normal Distribution.
  • 9. STANDARDISING A NORMAL DISTRIBUTION (CONT¡¯D) Example: Travel Time A survey of daily travel time had these results (in minutes):A survey of daily travel time had these results (in minutes): 26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34 The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes (you can copy and paste the values into the Standard Deviation Calculator if you want). Convert the values to z-scores ("standard scores"). To convert 26: ? = ?????.? ??.? = ?1.12; So 26 is ?1.12 Standard Deviations from the Mean Original Value Calculation Standard Score (z-score) 26 ?? ? ??. ? ??. ? = ?1.12 33 ?? ? ??. ? ??. ? = ?0.51 65 ?? ? ??. ? ??. ? = +2.30
  • 10. STANDARDISING A NORMAL DISTRIBUTION (CONT¡¯D) And here they are graphically: The rest of the scores can be calculated using the same formula. (You may try calculating those)
  • 11. STANDARDISING FORMULA The z-score formula that we have been using is: z is the "z-score" (Standard Score) x is the value to be standardized ¦Ì ('mu") is the mean ¦Ò ("sigma") is the standard deviation
  • 12. IN MORE DETAILS Here is the Standard Normal Distribution with percentages for every half of a standard deviation, and cumulative percentages:
  • 13. STANDARD NORMAL DISTRIBUTION TABLE Open the link below to access the table: https://www.mathsisfun.com/data/standard-normal-distribution- table.html Exercises Answer the questions at the bottom of this page: https://www.mathsisfun.com/data/standard-normal- distribution-table.html