AP®StatisticsCollege BoardRevision NotesUnit 1: Exploring One-Variable DataThe Normal DistributionFinding Proportions from Normal Distributions

Finding Proportions from Normal Distributions (College Board AP® Statistics)

Revision Note

Naomi C

Author

Naomi C

Expertise

Maths

Finding proportions from normal distributions

What is the standard normal table?

  • The proportion of a distribution of a variable X within a particular interval can be determined using the standard normal table given to you in the exam

  • The standard normal table shows the proportion of the distribution (or percentage of values) that lie below a particular z-score on a standard normal distribution

    • A z-score is the number of standard deviations that a value is from the mean

  • A z-score on the table is made up from

    • the row that corresponds to the correct units and tenths of the z-score

    • and the column that corresponds to the correct hundredths of the z-score

      • e.g. a z-score of -1.32 would use the row -1.3 and the column 0.02

Z-table section showing cumulative probabilities for z-values -1.4 to -1.2, and for decimals .00 to .03. Highlighted cell at z = -1.32, with value .0934.
A section of the standard normal table

  • You can also calculate proportions using your calculator

Exam Tip

When using technology to find the proportion of a distribution within a given interval it is essential to always show full working in Free Response Questions (FRQs). Write down any z-scores and parameters that you use.

How do I find a proportion below a given value using a standard normal table?

  • The proportion of a distribution of a variable X that lies below a given value is the area under the curve to the left of the given value

    • straight P open parentheses X less than a close parentheses equals p

A normal distribution curve with mean μ and shaded area to the left of a vertical dashed line at a. The mean is labeled μ and point a is labeled a.
Normal distribution with proportion of distribution less than the value 'a' shaded

  • To find this proportion using the standard normal table:

    • find the z-score for the value a

    • use this z-score to locate the appropriate row and column in the standard normal table

    • identify the cell that corresponds to this row and column

      • the value in this cell is straight P open parentheses Z less than z close parentheses, the proportion of the distribution that lies below the z-score

      • this is equal to straight P open parentheses X less than a close parentheses, the proportion of the distribution that lies below a

How do I find a proportion below a given value using a calculator?

  • You need a calculator that can calculate cumulative normal probabilities

  • You want to use the Normal Cumulative Distribution function

    • This is sometimes shortened to NCD, Normal CD or Normal Cdf

  • When using a calculator to find a proportion you can use the given values

    • You do not need to enter the associated z-scores first

      • However you should write down the z-score in the exam to support your work

  • To find straight P open parentheses X less than a close parentheses equals p, you will need to enter:

    • The lower bound

      • this needs to be a value that is sufficiently smaller than the mean

      • an easy option is to input lots of 9's for the lower bound with a negative sign (-99999999... or -1099)

    • The upper bound

      • this will be the value a

    • The mean, mu

    • The standard deviation, sigma

How do I find a proportion above a given value using a standard normal table?

  • The proportion of a a distribution of a variable X that lies above a given value is the area under the curve to the right of the given value

    • straight P open parentheses X greater than b close parentheses equals p

A normal distribution curve with mean μ and shaded area to the right of a vertical dashed line at b. The mean is labeled μ and point b is labeled b.
Normal distribution with proportion of distribution above b shaded

  • The total area under the curve is 1

    • So straight P open parentheses X greater than b close parentheses equals 1 minus straight P open parentheses X less than b close parentheses

  • To find this proportion using the standard normal table:

    • find the z-score for the value b

    • use this z-score to locate the appropriate row and column in the standard normal table

    • identify the cell that corresponds to this row and column

    • subtract the value in this cell from 1

      • the result is straight P open parentheses Z greater than z close parentheses, the proportion of the distribution that lies above the z-score

      • this is equal to straight P open parentheses X greater than b close parentheses, the proportion of the distribution that lies above b

How do I find a proportion above a given value using a calculator?

  • You will still use the Normal Cumulative Distribution function

  • To find straight P open parentheses X greater than b close parentheses equals p, you will need to enter:

    • The lower bound

      • this will be the value b

    • The upper bound

      • this needs to be a value that is sufficiently bigger than the mean

      • an easy option is to input lots of 9's for the lower bound with a positive sign (99999999... or 1099)

    • The mean, mu

    • The standard deviation, sigma

How do I find a proportion in a given range using a standard normal table?

  • The proportion of a distribution that lies in a given interval is the area under the curve between the lower bound value, a, and the upper bound value, b

    • straight P open parentheses a less than X less than b close parentheses equals p

A normal distribution curve with mean μ and shaded area to the right of a vertical dashed line at a and the left of a vertical dashed line at b.
Normal distribution with proportion of distribution between a and b shaded

  • To find this proportion using the standard normal table:

    • find the z-score for both the value a and the value b

    • use these z-scores in the standard normal table to find the proportion of the distribution that lies below each value

    • subtract the proportions of the two z-scores

      • straight P open parentheses Z less than z subscript b close parentheses minus straight P open parentheses Z less than z subscript a close parentheses

      • This is equal to straight P open parentheses X less than b close parentheses minus straight P open parentheses X less than a close parentheses,the proportion of the distribution that lies between a and b

How do I find a proportion in a given range using a calculator?

  • You will still use the Normal Cumulative Distribution function

  • To find straight P open parentheses a less than X less than b close parentheses equals p, you will need to enter:

    • The lower bound

      • this will be the value a

    • The upper bound

      • this will be the value b

    • The mean, mu

    • The standard deviation, sigma

Exam Tip

It is often useful to sketch a quick diagram of the section of the distribution that you are looking for.

Worked Example

An environmental group conducted a study measuring the dissolved oxygen content in a large number of samples of water to determine the water quality of a local lake. The results of the study found a mean level of 6.8 mg/l and a standard deviation of 0.7mg/l. A water sample with a dissolved oxygen content of less than 5 mg/l is considered to be unhealthy for fish.

What proportion of the water samples are considered to contain levels of dissolved oxygen that are unhealthy for fish?

Answer:

Method 1: Using the standard normal table

Define your variable and its distribution

Let X be the level of dissolved oxygen in the water samples

The distribution of X is normal with mean 6.8 and standard deviation 0.7

Draw a sketch describing the proportion of the distribution you are trying to find and write a probability statement for X

Normal distribution with mean 6.8 and an area shaded to the left of x=5. Dashed vertical lines at x=5 and x=6.8.

Calculate the z-score for the value of 5mg/l, using z equals fraction numerator x minus mu over denominator sigma end fraction

table row z equals cell fraction numerator 5 minus 6.8 over denominator 0.7 end fraction end cell row blank equals cell negative 2.5714... end cell end table

A value of 5mg/l lies 2.57 standard deviations below the mean

Using the z-score 2.57, find the row labeled -2.5 and the column labeled 0.7 on the standard normal table, and the cell where they intersect

Write a probability statement for Z

straight P open parentheses Z less than negative 2.57 close parentheses equals 0.0051

Explain the value in the context of the question, leaving the value from the table to 4 significant figures

The proportion 0.0051 (or 0.51%) of water samples taken contain an unhealthy dissolved oxygen level of less than 5mg/l

Method 2: Using a calculator

Define your variable and its distribution

Let X be the level of dissolved oxygen in the water samples

The distribution of X is normal with mean 6.8 and standard deviation 0.7

Draw a sketch describing the proportion of the distribution you are trying to find and write a probability statement for X

Normal distribution with mean 6.8 and an area shaded to the left of x=5. Dashed vertical lines at x=5 and x=6.8.

Even though you do not need to enter it into your calculator, it is a good idea to write down the calculation of the z-score for the value of 5mg/l, using z equals fraction numerator x minus mu over denominator sigma end fraction,

table row z equals cell fraction numerator 5 minus 6.8 over denominator 0.7 end fraction end cell row blank equals cell negative 2.5714... end cell end table

Write a probability statement for Z

straight P open parentheses Z less than negative 2.57 close parentheses

Write down the parameters for the situation

lower bound = -1099

upper bound = 5

mu equals 6.8

sigma equals 0.7

Enter these values into the Normal Cumulative Distribution function on your calculator

straight P open parentheses X less than 5 close parentheses equals 0.0051

Explain the value in the context of the question

The proportion 0.0051 (or 0.51%) of water samples taken contain an unhealthy dissolved oxygen level of less than 5mg/l

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    Naomi C

    Author: Naomi C

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.