Chapter 5

A sample with a mean of M = 50 and a standard deviation of s = 12 is being transformed into z-scores. After the transformation, what is the standard deviation for the sample of z-scores?

1

2 purposes of z-scores

1) Tell exactly where within a distribution the original scores are located 2) Standardize an entire distribution E.g., Distribution of IQ scores Standardized so that they have a mean of 100 and a standard deviation of 15 Allows you to understand and compare IQ scores even though they come from different tests Takes different distributions and makes them equivalent

The procedure for standardizing a distribution to create new values for the mean and standard deviation is a two-step process that can be used with a population or a sample:

1) The original scores are transformed to z-scores 2) The z-scores are then transformed into new X values so that the specific mean and standard deviation are attained

Two components of a z-score

1) The sign tells whether the score is located above (+) or below (-) the mean, and 2) The number tells the distance between the score and the mean in terms of the number of standard deviation

E.g., mu = 100, sigma = 10 What z-score corresponds to a score of X = 130 in this distribution?

130-100 = 30 / 10 = 3

Maria's z-score of z = +0.50 indicates that she is located above the mean by 1/2 a standard deviation In the new, standardized deviation this location corresponds to X = ____ Joe's z-score of z = -1.00 indicates that he is located below the mean by exactly 1 standard deviation. In the new distribution, this location corresponds to X = ____

55 40

In a population with mu = 60, a score of X = 58 corresponds to a z-score of z = -0.50. What is the population standard deviation?

58 - 60 = -2 / -.5 = 4

For a population with mu = 80 and sigma = 12, what is the z-score corresponding to X = 92

92-80 = 12/12 = 1

E.g., m = 86, s = 7 What z-score corresponds to a score of X = 95 in this distribution?

95 - 86 = 9 / 7 = 1.2857

Example: Does hormone have an effect on weight? Distribution of weights for population of adult rats A hormone-injected rat weighs X = 418 grams vs X = 450 grams

A hormone-injected rat weighs X = 418 grams More than the average non-treated rat (i.e., 400 grams), but is this convincing evidence the hormone has had an effect? Z-score = 0.90 418-400 = 18/20 = 0.9 No, injected rat still looks the same as a regular, non-treated rat Conclusion: hormone does NOT appear to have an effect on weight A hormone injected rat weights X = 450 grams Z-score = 2.50 450-400 = 50/20 = 2.5 Hormone-injected rat is substantially bigger than most ordinary rats Conclusion: hormone DOES have an effect on weight

E.g., mu = 100, sigma = 15, z = + 2.00

Above the mean (+) by a distance of 2 standard deviations (30 points)

What location in a distribution corresponds to z = -.00?

Below the mean by a distance equal to 2 standard deviations

E.g., Dave received a score of X = 60 on a psychology exam and a score of X = 56 on a biology exam For which course should David expect a better grade Scores are from two different distributions, you cannot make any direction comparison Before you can begin to make comparisons, you must know the value for mean and standard deviation for each distribution

Biology mu = 48, sigma = 4 Psychology mu = 50 ssigma = 10 Compute the two z-scores to find the two locations: Biology Z = 56-48 = 8 /4 = 2 Psychology 60-50 = 10/10 = 1 In terms of relative class standing, Dave is doing much better in the Biology class

In a population of scores, X = 45 corresponds to z = +2.00 and X = 30 corresponds to z = -1.00. What is the population mean?

Distance between X = 30 and X = 45 is 15 Distance between z-scores is 3 3 standard deviations = 15 1 standard deviation = 5 X = 30 is 1 standard deviation below mean 30 + 5 = 35

E.g., in a population distribution, a score of X = 54 corresponds to z = +2.00 and a score of X = 42 corresponds to z = -1.00. What are the values for the mean and standard deviation for the population?

Distance between X = 42 to X = 54 is 12 points X = 42 is located 1 standard deviation below the mean, X = 54 is located 2 standard deviations above the mean 3 standard deviations apart 3s = 12 points s = 4 points X = 42 is 1 standard deviation below the mean Mean = 42 + 4 = 46

In a sample, X = 70 corresponds to z = +2.00 and X = 65 corresponds to z = +1.00. What are the sample mean and standard deviation?

Distance between X = 65 and X = 70 is 5 Distance between z-scores is 1 S = 5 65 - 5 = 60 = M

For the past 20 years, the high temperature on April 15th has average mu = 62 degrees with a standard deviation of sigma = 4. Last year, the high temperature was 72 degrees. Based on this information, last year's temperature on April 15th was

Far above average

____ statistics are techniques that use the information from samples to answer questions about populations

Inferential

Advantage of standardizing distributions?

It makes it possible to compare different scores or different individuals even though they come from different distributions Normally, if 2 scores came from different distributions, it would be impossible to make any direction comparisons between them

Last week Sarah had exams in math and in Spanish. On the math exam, the mean was mu= 30 with sigma = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was mu = 60 with sigma = 6 and Sarah had a score of X = 65. For which class should Sarah expect the better grade?

Math m = 30 s =5 Score of X = 45 Z = 45-30 = 15/5 = 3 Spanish m = 60 s = 6 Score of X = 65 65-60 = 5/6 = 0.8333 Math

*Transforming a distribution of X values to z values does ____ move scores from one position to another; the procedure simply ____ each score

NOT relabels

Using z-scores, a sample with M = 37 and s = 6 is standardized so that the new mean is M = 50 and s = 10. How does an individual's z-score in the new distribution compare with his/her z-score in the original sample?

New z = old z

____ will give a rational for deciding exactly where to set z-score boundaries for "representative" vs. "extreme"

Probability

Some people find z-scores cumbersome because they contain negative values and decimals Solution?

Standardize a distribution by transforming scores into a new distribution with predetermined mean and standard deviation that are positive whole numbers

z-score formula for a population

The numerator X - mu is a deviation score The deviation score is divided by sigma because we want the z-score to measure distance in terms of standard deviation units

If all of the scores in a sample are transformed into z-scores, the result is a sample distribution of z-scores Will have the same properties that exist when a population of X values is transformed into z-sores Which are?

The distribution of the sample of z-scores will have the same shape as the original sample of scores The sample of z-scores will have a mean of M = 0 The sample of z-scores will have a standard deviation of s = 1

If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties Standard deviation

The distribution of z-sores will always have a standard deviation of 1

If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties Mean

The z-score distribution will always have a mean of zero (i.e., The original population mean is transformed into a value of zero in the z-score distribution) A convenient reference point

E.g., You received a score of X = 76 on the statistics exam How did you do? Mean is 70 At this point, you know that your score is 6 points above the mean, but you still do not know exactly where it is located

Two possible distributions Standard deviation 3 Score of 76 is at the extreme right-hand tail, one of the highest scores in the distribution Standard deviation = 12 Score of 76 is only slightly above average

The definition and purpose of a z-score is the same for a sample as a population What's the difference?

Use the sample mean Use the sample standard deviation

Under what circumstances would a score that is 15 points above the mean be considered an extreme score?

When the standard deviation is much smaller than 15

If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties Shape

Will have the same shape as the original distribution of scores

A distribution with mu = 35, and sigma = 8 is being standardized so that the new mean and standard deviation will be mu = 50 and sigma = 10. When the distribution is standardized, what value will be obtained for a score of X = 39 from the original distribution?

X = 39 Z = 39 - 35 = 4/8 50 + 5 = 55 55

A set of scores has a mean of mu = 62 and standard deviation of sigma = 8. If the scores are standardized so that the new distribution has a mu = 50 and sigma = 10, what new value would be obtained for a score of X = 59 from the original distribution?

X = 59 Z = 59 - 63 = -4/8 = -0.5 50 - 5 = 45 45

In a sample with a standard deviation of s = 8, a score of X = 64 corresponds to z = -0.50. What is the sample mean?

X = 64 is located .5 standard deviation below the mean S = 8 Mean = 64 + 4 = 68

For a sample with M = 72 and s = 4, what is the X value corresponding to z = -2.00

X = M + zs X = 72+ (-2)(4) X = 64

E.g., for a distribution with a mu of 60 and sigma of 8, what X value corresponds to a z-score of z = -1.50?

X is below the mean by a distance equal to 1.5 standard deviations 1.5 * 8 points = 12 points 60 - 12 = 48

E.g., In a sample with a standard deviation of s = 6, a score of X = 33 corresponds to z = +1.50. What is the mean for the sample?

Z-score of +1.50 indicates that the score is located above the mean by a distance of 1.5 standard deviations With a standard deviation of s = 6, the distance is (1.5)(6) = 9 Score is located 9 points above the mean The score is X = 33, so the mean must be 24

E.g., In a population with a mean of m = 65, a score of X = 59 corresponds to z = -2.00. What is the standard deviation for the population?

Z-score of -2.00 indicates the score is located below the mean by a distance of 2 standard deviations X = 59 is located below the mean by a distance of 6 points 2 standard deviations is a distance of 6 points 1 standard deviation must be 3 points

A score of X = 73 was obtained from a population. Which set of population parameters would make X = 73 an extreme and unrepresentative score? a) mu = 65 and sigma = 8 b) mu = 65 and sigma = 3 c ) mu = 70 and sigma = 8 d) mu = 70 and sigma = 3

b) mu = 65 and sigma = 3

In general, it is easier to use the ____ of a z-score, rather than a formula, when you are changing z-sores into X values

definition

A population with mu = 80 and sigma = 15 is transformed into z-scores. After the transformation, what is the mean for the population of z-scores?

mu = 0

A score by itself does not necessarily provide much information about its position within a distribution Original, unchanged scores that are the direct result of measurement are called ____

raw scores

The advantage of having a standard deviation of 1 is that the numerical value of a z-score is exactly the ___ as the number of standard deviations from the mean

same E.g., a z-score of z = 1.50 is exactly 1.50 standard deviations from the mean

Location of score within the distribution depends on the ____ as well as the ____

standard deviation mean

When any distribution (with any mean or standard deviation) is transformed into z-scores, the resulting distribution will always have a mean of m = 0 and a standard deviation s = 1 Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a ____

standardized distribution

A z-score establishes a relationship between ...

the score, the mean, and the standard deviation *This relationship can be used to answer a variety of different questions about scores and the distributions in which they are located

What is the purpose of z-scores or standard scores?

to identify and describe the exact location of each score in a distribution

To make raw scores more meaningful, they are often ____ into new values that contain more information

transformed

Z-score near 0 would be considered a fairly ____ or representative individual Z-score beyond +2.00 and -2.00 would be considered ____ from most individuals in the population

typical "noticeably different"

Interpretation of research result depends on whether the sample is noticeably different from the population One technique for deciding whether a sample is noticeably different is to use ____

z-scores

Standard score

zscore