Foundations 12 Section 5.5 Z-Scores

Z-ScoresZ-Score: a standardized value that indicates the number of standard deviations ( ) that a data value ( x ) is away from the mean ( ). It indicates the position of the raw data value on a standard normal distribution.

The standard normal curve has a mean of 0 and a standard deviation of 1.

The numbers on the x-axis are the z scores.

If a point has a z-score of 1.5, this means that the data point is 1.5 standard deviations above the mean.

Properties of Z-Scores:1. A z-score for a data value describes the number of standard deviation above or below the mean.2. A negative z-score indicates that the value is below the mean and is shown to the left of the mean on the

standard normal distribution curve.3. A positive z-score indicates that the value is above the mean and is shown to the right of the mean on the

standard normal distribution curve.4. A z-score table give the percentage of data at or to the left of the z-score.5. The mean, median and mode have a z-score of 0.

Z Tables

Finding a z score for a point allows us to determine the percentage of values found above or below that point by using the normal distribution curve or by using a z-score table which provides a quick reference to percentages for z-scores.

Z- score table: a table that displays the fraction of data with a z-score that is less than any given data value in a standard normal distribution.

For example, if a data point has a z-score of 1.0, it means that it is 1 standard deviation above the mean and 84% of the data points are lower than this point. This percentage can be determined by referring back to the normal curve diagram below (50% + 34%) or by using the z-score table which gives a value of 0.8413 or approximately 84% for a z-score of 1.

Foundations 12 Section 5.5 Z-Scores

Finding Areas Under the Standard Normal Curve1. To find the area to the left of z, find the area that corresponds to z in the Standard Normal Table.

If z = 1.23, determine the percent of the data to the left.

89.07 % of the data is found to the left of a z score of 1.23.

2. To find the area to the right of z, use the Standard Normal Table to find the area that corresponds to the z value. Then subtract the area from 1.If z =1.23, determine the percent of the data to the right.

10.93 % of the data falls to the right of a z-score of 1.23.

Foundations 12 Section 5.5 Z-Scores

To find the area between two z-scores, find the area corresponding to each z-score in the Standard Normal Table. Then subtract the smaller area from the larger area.Determine the percent of the data between a z-score of 1.2 and -0.75.

66.41% of the data falls between a z-score of 1.23 and a z score of 0.75.

Example 1: Understanding Z-Scores

Use the z-table to calculate the percentage of data represented by the given z-score.

a. -1.96 b. 1.5

c. -2.75 < z < 2.75 d. -1.5 < z < 1.03

Example 2: Understanding the Z-Score table

Use the table to find the z-score that gives the percentage of data shown.

a.

b. d.

Foundations 12 Section 5.5 Z-Scores

What happens when we don’t have a standard normal distribution?

The z-score for a particular data value is the number of standard deviations the data is above or below the mean.

Example 3: Calculating and Displaying z-scores on the Normal Curve

Consider a normally distributed set of data with a mean value of 12 and a standard deviation of 2. Complete the following table.

Data Value Normal Distribution

# of standard deviations

Above or below

Corresponding z-score

18

8

12

Foundations 12 Section 5.5 Z-Scores

15

Label a normal distribution graph with the following information: mean 19.2 and a standard deviation of 3.6

given data:

z-scores:

Z-score Formula

A z-score for a data value can be calculated using the formula

where x is a particular data value, is the mean, and is the standard deviation

A positive z-score indicates the data value lies above the mean, while a negative z-score indicates the data value lies below the mean.

Given the mean and standard deviation for a population, one can determine the z-score and the percentage of values above and below that point.

Example 4: Computing a Z-Score

IQ tests are used to measure a person’s intellectual capacity at a given time. IQ test results are normally distributed with a mean of 100 and a standard deviation of 15. If a person’s scores 119 on an IQ test, how does this score compare with the rest of the population?

.

Example 5: Comparing Z-Scores from 2 different Normal Distributions

One can compare data values from different normal distributions, with different means and standard deviations, by converting the data values into z-scores.

Haley recently completed two exams, one in Math and the other in English. She recorded the following scores:

Foundations 12 Section 5.5 Z-Scores

Math 70% (class mean was 55 and the standard deviation was 10) English 60% (class mean was 50 and the standard deviation was 5)

Which exam mark represents the better performance (relative to the class)? Explain.

Math English

Example 6: Using Z –Scores To Determine Data Values

Athletes should replace their running shoes before the shoes lose their ability to absorb shock. Running shoes lose their shock-absorption after a mean distance of 640 km, and a standard deviation of 160 km. Andrew is an elite runner and wants to replace his shoes at a distance when only 25% of people would replace their shoes. At what distance should he replace his shoes?

Using a z-score table:

Sketch the normal curve. The z-score for 25 % of the data is needed.

Now refer to the z-score tables for a value close to 0.25

A z-score that represents an area of 0.25 is halfway between ____ and ___ , or about ___.

Therefore, Andrew should replace his running shoes after____ km.

Example 7: Solving A Quality Control Problem

Morrison’s Outdoor Equipment produces bungee cords. When the manufacturing process is running well, the lengths of the bungee cords produced are normally distributed, with a mean of 45.2 cm and a standard deviation of 1.3 cm. Bungee cords that are shorter than 42.0 cm or longer than 48. 0 cm are rejected by the quality control workers.Using z-scores tables

a. If 20 000 bungee cords are manufactured each day, how many bungee cores would you expect the quality control workers to reject?

Foundations 12 Section 5.5 Z-Scores

Determine the z-scores for the minimum and maximum length

The area under the curve to the left of_____ represents the percent of rejected bungee cords less than 42 cm. The area under the curve to the right of _____ represents the percent of rejected bungee cords greater than 48 cm.

Look up the z-score in the tables.

Area to the left of -2.46 = Area to the right of 2.15 =

Percent rejected = area to the left of -2.46 + area to the right of 2.15

Percent rejected =

Total number rejected =

b. What action might the company take as a result of these findings?

Example 8: Determining Warranty Periods

A manufacturer of personal music players has determined that the mean life of the player is 32.4 months, with a standard deviation of 6.3 months. What length of warranty should be offered if the manufacturer wants to restrict repairs to less than 1.5% of all the players sold?

Determine the z-score that corresponds to an area under the normal curve of 0.015

Substitute the known values into the z-score formula and solve for x.

The manufacturer should offer a warranty of _____months (manufacturer wants to round down)

Foundations 12 Section 5.5 Z-Scores