Prob and Stat

6.1 and 6.2

Normal Curve

  • Normal curve is the symmetrical about the mean μ.
  • The mean is at the middle and divides the area into halves.
  • The total area under the curve is equal to 1
  • It is completely determined by its mean and standard deviation σ

μ - Sample Mean σ - Standard Deviation

Calculating Standard Deviation:

  • 1. Work out the Mean (the simple average of the numbers)
  • 2. Then for each number: subtract the Mean and square the result
  • 3. Then work out the mean of those squared differences.
  • 4. Take the square root of that and we are done!

What is the Empirical Rule?

The empirical rule states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean.


The empirical rule can be broken down into three parts:

  • 68% of data falls within the first standard deviation from the mean.
  • 95% fall within two standard deviations.
  • 99.7% fall within three standard deviations.

Z Score

A z-score (aka, a standard score) indicates how many standard deviations an element is from the mean. A z-score can be calculated from the following formula. z = (X - μ) / σ where z is the z-score, X is the value of the element, μ is the population mean, and σ is the standard deviation.