Lecture 14 & 15 - z-scores

z-score

a standardized measure of how far an individual score is away from the mean in terms of the standard deviation of the distribution

z-scores are for when

• you want to compare some score to the mean in standard units

• you know the population mean and standard deviation

T/F: a z-score equal to 0 represents an element equal to the mean

true

T/F: a z-score equal to 1 represents an element that is 1 standard deviation greater than the mean

true

T/F: a z-score equal to 2 represents an element that is 2 standard deviations greater than the mean

true

T/F: a z-score equal to -1 represents an element that is 1 standard deviation greater than the mean

false

T/F: a z-score equal to -2 represents an element that is 2 standard deviations greater than the mean

false

Standard normal curve

a theoretical normal distribution with a mean of 0 and a SD of 1

The standard normal

one set of parameters for a normal distribution

z-scores are unitless and help us understand our data generally which means?

• z-scores thus talk about data in terms of distance in SDs from the mean; we ignore the initial units

• z-score of 1 always means 1 SD above the mean

To compute a z-score, you need

• the score you want to convert to a z-score

• the mean of the distribution

• the standard deviation of the distribution

Standard curve gives us easy access to information about what?

the proportion under the curve

How does a z-score table help you?

a z-score table lets you convert from z-scores to proportions

z-scores are listen along with what two values?

column B and column C

Column B

proportion of data between the mean and the z-score

Column C

proportion of data beyond the z-score

If we care about what proportion of data is above a z-score, we care about what column?

column C

Determining what you need from z-table

• find the z-score in column A

• look at columns B and C

• determine which value you need

To find amount below a negative z-score using the tables

• find the positive z-score in Column A (aka use the absolute value)

• Column B is amount between mean and this z-score

• Column C is amount more extreme than this z-score

  • This works because the distribution is symmetric

What if we want to know area above a negative z-score

• find the absolute value of z-score in Column A

• add .5 to column B

Why would you want to calculate a z-score?

• you want to know how much better or worse a score is than average considering how variable the scores are

• you want to compare scores across two different distributions in a standardized format

What do you need to calculate a z-score?

• a population mean

• a raw score from that population

• the standard deviation of the population

  • there are other things you can calculate with only two of these

Interpreting z-scores

• a positive z-score means that the value is above the mean

• a negative z-score means that the value is below the mean

• a z-score of 0 means the score is identical to the mean

For a normal curve…

The area under the curve in our plotted distribution always add up to 100% of our data

• 68% falls within +/- 1 standard deviation of the mean

• 95% fall within +/- 1.96 SDs

Z-scores apply our values to the standard normal curve

special case of a normal curve with a distribution with a mean of 0 and a standard deviation of 1