Probability+topic+page

=SIMULATIONS (separate page - click to view)=

=Examples for Achieved, Merit, Excellence= (click to download) [|Sample Worked Questions for NORMAL DISTRIBUTION.doc] =Normal Distribution Notes=

This is common in many naturally-occuring forms of data e.g. heights, weights. The centre of the distribution is the mean AND the median. The shape is symmetrical. e.g. if it represented HEIGHTS, this distribution shows that the majority of people are an "average" height, extremely short or extremely tall is less common.



The **STANDARD normal** is the same diagram with all values expressed as standard deviations from the mean. The mean is automatically zero (zero standard devitations away from itself!)

__Using the Normal Distribution table for probabilities.__
Z scores are the NUMBER of standard deviations from the mean. When you look a z-score up on a table, the probability it gives the probability of being between the mean and that number (shaded in the centre).

e.g. z-score of 1.314 gives probability of 0.4055

The shaded range has a probability of 0.4055 (it's in the centre, so use the result straight from the table.

The shaded range is 0.4055 + 0.5 (number from table, plus a whole half is shaded as well)



0.4055 is the area between 0 and 1.314. This is not what is shaded. You need to subtract this value from 0.5 (whole top half minus unshaded section below 1.314) Probability = 0.5 - 0.4055 = 0.0945

This is trickier. Look up both z-scores (1.8 and 1.314). Subtract the two probabilities to get the probability of being between them. z score of 1.8 gives P = 0.4641 z score of 1.314 gives P = 0.4055

Final probability = 0.4641 - 0.4055 = 0.0586

For this question, you have two shaded areas near the centre. It is as simple as looking up both z scores and adding their probabilities. z score of 0.823 gives P = 0.2947 z score of 1.314 gives P = 0.4055

Total probability = 0.2947 + 0.4055 = 0.7002

__Calculating a z-score__ You are given this formula: [[image:z_score_formula.png]] but you must know what the symbols mean!
Remember that X is the number you want to convert, subtract the mean, divide the answer by the size of the standard deviation. **z-scores should be rounded to 3dp, as that is the limit of the tables.**

=Achieved Questions=


 * You will be given the mean and the standard deviation. You will be asked for the probability of a certain range.
 * You must convert all values into z-scores
 * It's a good idea to sketch two diagrams: one with the original numbers on it, and the other with the numbers converted to z-scores (standard normal distribution).
 * Use your table to look up/calculate the probability. CHECK THE LOCATION OF THE SHADING!