Percentages and simple Probability

Percentages and Probabilities.

• The corresponding percentages from the previous slide can be expressed as probabilities.
  34.13% = 34.13/100 = .3413
  13.59% = 13.59/100 = .1359
  02.15% = 02.15/100 = .0215
  
• The proportion of scores between any two Z-scores is the same as the probability of selecting another score between those two scores.
• Keep in mind, the complete table of z-scores and their associated probabilities is available at multiple sites online.
  
  http://www.mathsisfun.com/data/standard-normal-distribution-table.html

The BOTTOM LINE!

Knowing the formula for converting a raw score to a z-score and knowing the percentages of the standard normal curve (or at least having access to a table):

• We can calculate the percentage of scores between any two points of the normal distribution.
• For instance, between a raw score and the mean or between any two Z-scores.

More on Percentages.

• When figuring percentage of scores between any two z-scores or between the mean and a Z-score, pay close attention to the column in the table you need.

Z	% Mean to Z	% in Tail
.00	.00	50.00
.01	.40	49.60
.	.	.
.	.	.
.	.	.
.50	19.15	30.85
1.00	34.13	15.87

http://davidmlane.com/hyperstat/z_table.html

Percentiles = Directional Percentages

• Remember, when finding a percentile rank, the area under the normal curve becomes ranked (from 1 to 100 percentiles).
• For instance, we know that a Z-score of +2 represents 2 standard deviations above the mean. This same location, when converted to a percentile would be the 98th percentile.
  • Explanation: 50% of the scores are below the mean, (+) 34% of the scores between the mean and a z-score of +1, (+) the other 14% of scores between a Z-score of +1 and a Z-score of +2 = 98% of the scores, or the 98th percentile.

Percentiles

• Likewise, a Z-score of -1 which is one standard deviation below the mean would be expressed as the 16th percentile.
  • Explanation: 2% of the scores are beyond 2 standard deviations below the mean, (+) 14% of the scores between 2 standard deviations below the mean and 1 standard deviation below the mean = 16% of the scores are below our Z-score of -1; a raw score with the Z-score of -1 is the 16th percentile.

Z-scores along the bottom, probabilities around the top
Memorize this image...burn it into your brain!!

Percentages & simple Probabilities

• The numbers on the outside of the standard normal curve from the previous slide can be expressed as percentages or probabilities.
  34% = .3413
  14% = .1359
  02% = .0215
  
• The proportion of scores between any two Z-scores is the same as the probability of selecting a case between those two Z-scores.

More on simple Probability

• The range of probabilities: Zero to One.
  • Range =
• Expressed as an italicized, lower-case p
  • If there is no chance the event will occur, then p = 0
  • If the event is certain to occur, then p = 1
  • p means the probability of the event is less than .02
• Calculating simple probability.
  • The number of possible successful or desirable outcomes divided by the number of all possible outcomes.

Interpretation of Probability

• Long-run relative-frequency interpretation.
  • What you would expect to get, in the long run, if you were to repeat the experiment many times.
  • The probability of rolling a 5 on a six-sided die is: p = .166, so if we roll the die 1000 times, we would expect to get a 5 roughly 166.6 times.
    • This is the interpretation we typically use.
• Subjective interpretation of probability.
  • How certain one is that a particular event will happen.
    • There is a 16.66% chance I'll roll a 5 on one independent roll of the die.

Example simple Probabilities

Remember, simple probability is the number of desired outcomes divided by the number of possible outcomes.

• Probability of getting a head or a tail on one toss of the coin?
  p = 1
  
• Probability of getting a head on one toss of the coin?
  p = .50
  
• Probability of NOT getting a head or a tail on one toss of the coin?
  p = 0
  
• Probability of getting a head on one toss of the coin after 2500 tosses of the coin?
  p = .50
  
• Remember the old saying: ``The coin has no memory.''

Practice Examples

• The `Bear Cubs Problem'
• The `Balls of Two Colors' problem

Available toward the bottom of the page at:

http://www.cut-the-knot.org/probability.shtml