Tails on a Normal Distribution (College Board AP® Statistics)

Tails on a normal distribution

What are tails on a normal distribution?

• Tails are the regions under the normal curve on the extreme left- or right-hand sides of the distribution

How do I find the most extreme P% of values from a normal distribution?

• The most extreme P% of values in a normal distribution lie in the left-hand tail and the right-hand tail

  • Both tails are required

• As the normal distribution is symmetric, this means

  • of P% of values lie in the left-hand tail

  • and of P% of values lie in the right-hand tail

How do I find the middle Q% of values from a normal distribution?

• To find the middle Q% of values from a normal distribution

  • it is easier to work out the percentage of values you do not want

    • 100% - Q%

  • then halve this value to give the percentage on each tail

    • %

• For example, the middle 90% of values has

  • 10% across both tails

  • so 5% on each tail

  • If the boundary values on the standard normal distribution are then

    • there is a total of 5% + 90% = 95% to the left of

    • so

How do I use the t-distribution tables to find boundary values?

• The usual way to find boundary values for the middle 90% is to

  • let the positive boundary value be , then solve

    • This is an inverse normal calculation

    • i.e. find the probability closest to 0.95 in the normal tables

    • then read off the z-score

• However, for certain common percentages that come up a lot (e.g the middle 50%, 90%, 95%, 99% etc of values) it is easier and more accurate to use

  • the very last row of the t-distribution table ('t-table')

    • called row infinity,

  • as row infinity is a row of z-scores from the standard normal distribution

• The percentage shown underneath row infinity is the middle percentage of values

  • not the percentage to the left

  • e.g. the middle 90% of values has a positive boundary z-score of 1.645 (from the t-tables)

    • This is more accurate than '1.64 or 1.65' from the normal tables