NORMAL DISTRIBUTION

• Normal Curve


  Bell-shaped, smooth, mathematically defined curve that is highest at its center
• Normal Distribution


  • Symmetrical
  • Unimodal
  • Asymptotical
  • Mean, Median, & Mode are Equal

  
• 50% of the scores occur above the mean and 50% of the scores occur below the mean
• Approximately 34% of all scores occur between the mean and +1 standard deviation; 34% of all scores occur between the mean and -1 standard deviation
• Approximately 68% of all scores occur between +1 and -1 standard deviation
• Approximately 95% of all scores occur between the +2 and -2 standard deviations
• Tails


  The area between ±2 and ±3 standard deviation
• Skewed Distribution


  Strong tendency for the mean, median, and mode to be located in predictably different positions

  
• Positively Skewed


  • The peak (highest frequency) is on the left-hand side. The most likely order of the three measures of central tendency from smallest to largest (left to right) is the mode, the median, and the mean

  
• Negatively Skewed


  • The scores piling up on the right-hand side and the tail tapering off to the left. The most probable order for the three measures of central tendency from smallest to largest (left to right), is the mean, the median, and the mode
• Standard Scores


  Raw score that has been converted from one scale to another scale, where the latter scale has some arbitrarily set mean and standard deviation
• Purpose of Score Conversion

  • Each standard score tells the exact location of the original X value within the distribution
  • The standard score form a standardized distribution that can be directly compared to other distributions that also have been transformed into standard scores
• Types of Standard Scores


  • z Scores
  • t Scores
  • Stanine
  • Sten
  • Percentile Rank
  • IQ Scores
• z Score


  • Specifies the precise location of each X value within a distribution
  • The sign of the z-score (– or +) signifies whether the score is above the mean (positive) or below the mean (negative)
  • The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and mean

  
• t Score


  • Standard score system composed of a scale that ranges from 5 standard deviations below the mean to 5 standard deviations above the mean
  • z score can be used to get the T score
  • Scale: fifty plus or minus ten scale, μ = 50 and σ = ± 10
• Stanine


  • Score with a mean of 5 and a standard deviation of approximately 2 and divided into nine units
  • z score can be used to get the stanine
  • Scale: 5 plus or minus 2 scale, μ = 5 and σ = ± 2
• Sten


  • Score with a mean of 5.5 and a standard deviation of approximately 2 and divided into nine units
  • z score can be used to get the sten
  • Scale: 5.5 plus or minus 2 scale, μ = 5.5 and σ = ± 2
• Percentile Rank


  • Measurement that shows the percentage of scores within a norm group that is lower than the score you're measuring
  • May not denote an actual test score or other assessment score; it only represents an item's rank against a larger group's places between 0 and 100
• IQ Score


  • Standardized score used by many intelligence tests; also called deviation intelligence quotient
  • z score can be used to get the IQ score
  • Scale: 100 plus or minus 15 scale, μ = 100 and σ = ± 15
• Probability


  Used to predict the type of samples that are likely to be obtained from the population
• Area Under the Normal Curve

  • The standard normal distribution is a probability distribution, so the area under the curve between two points tells you the probability of variables taking on a range of values
  • The total area under the curve is 1 or 100%

  
• Normal distribution


  Population distribution with a mean (μ) and standard deviation (σ)
• The population distribution of Math SAT scores is normal with a mean of μ = 500 and a standard deviation of σ = 100
• Probability of selecting an individual with Math SAT score greater than 700

  p(X > 700)
• Calculating probability of X > 700

  1. Compute z-score

  2. Restate probability notation

  3. Use area under normal distribution

  
• Probability of selecting someone with Math SAT score greater than 700 is 2.28%
• Probability of selecting an individual with Math SAT score greater than 600


  p(X > 600)
• Calculating probability of X > 600

  1. Compute z-score

  2. Restate probability notation

  3. Use area under normal distribution

  
• Probability of selecting someone with Math SAT score greater than 600 is 15.87%
• Probability of selecting an individual with Math SAT score less than 400


  p(X < 400)
• Calculating probability of X < 400

  1. Compute z-score

  2. Restate probability notation

  3. Use area under normal distribution

  
• Probability of selecting someone with Math SAT score less than 400 is 15.87%
• Probability of selecting an individual with Math SAT score less than 700

  p(X < 700)
• Calculating probability of X < 700

  1. Compute z-score

  2. Restate probability notation

  3. Use area under normal distribution

  
• Probability of selecting someone with Math SAT score less than 700 is 97.72%
• Probability of selecting an individual with Math SAT score greater than 400


  p(X > 400)
• Calculating probability of X > 400

  1. Compute z-score

  2. Restate probability notation

  3. Use area under normal distribution

  
• Probability of selecting someone with Math SAT score greater than 400 is 84.13%
• Probability of selecting an individual with Math SAT score between 400 and 700

  p(400 < X < 700)
• Calculating probability of 400 < X < 700

  1. Compute z-scores

  2. Restate probability notation

  3. Use area under normal distribution