# Basic Facts About z-Scores

October 22, 2008 by kltangen

Filed under Basic Facts

Laws have accuracy beyond doubt.

Comparing self to self

Comparing self to a standard

Comparing self to a group

5 z-score applications

1. Describe the location of an individual score

2. Use as a cutoff score

3. Convert a distribution to a standardized distribution

4. Linearly transform a distribution

5. Compare distributions with different means and standard deviations.

NOW YOU CHOOSE:

Day 4: z-Score

A Bit More About z-Scores

Even More About z-Scores

How To Calculate z-Scores

Practice Problems

Basic Facts About z-Scores

Vocabulary

Formulas For z-Scores

Quiz 4

Summary

# Formula For z-Score

Day 4: z-Score

A Bit More About z-Scores

Even More About z-Scores

How To Calculate z-Scores

Practice Problems

Basic Facts About z-Scores

Vocabulary

Formulas For z-Scores

Quiz 4

Summary

# Practice Problem Answers: z-scores

October 22, 2008 by kltangen

Filed under Calc Solutions

Item 1

Calculate the z-score, assuming:

X = 10

mean = 50

stdev = 10

X – mean = – 40

z score = **–** 4.0

Item 2

Calculate the z-score, assuming:

X = 80

mean = 50

stdev = 15

X – mean = 30

z score = 2.0

Item 3

Calculate the z-score, assuming:

X = 112

mean = 100

stdev = 15

X – mean = 12

z score =** **.80

Item 4

Calculate the raw score, assuming:

z = 1.5

mean = 115

stdev = 10

z * stdev = 15

X = 130** **

Item 5

Calculate the raw score, assuming:

z = -1.37

mean = 100

stdev = 20

z * stdev = 27.4

X = 72.6

Item 6

Calculate the raw score, assuming:

z = 2.54

mean = 80

stdev = 20

X = 130.8

2.54*20 = 50.8

50.8 + 80 = 130.8

Day 4: z-Score

A Bit More About z-Scores

Even More About z-Scores

How To Calculate z-Scores

Practice Problems

Basic Facts About z-Scores

Vocabulary

Formulas For z-Scores

Quiz 4

Summary

# Practice Problems: z-Score

October 22, 2008 by kltangen

Filed under Practice Problems, z-scores

Item 1

Calculate the z-score, assuming:

X = 10

mean = 50

stdev = 10

X – mean _____

z score _____

Item 2

Calculate the z-score, assuming:

X = 80

mean = 50

stdev = 15

X – mean _____

z score _____

Item 3

Calculate the z-score, assuming:

X = 112

mean = 100

stdev = 15

X – mean _____

z score _____

Item 4

Calculate the raw score, assuming:

z = 1.5

mean = 115

stdev = 10

z * stdev _____

X _____

Item 5

Calculate the raw score, assuming:

z = -1.37

mean = 100

stdev = 20

z * stdev _____

X _____

Item 6

Calculate the raw score, assuming:

z = -1.37

mean = 100

stdev = 20

X _____

NOW YOU CHOOSE:

Day 4: z-Score

A Bit More About z-Scores

Even More About z-Scores

How To Calculate z-Scores

Practice Problems

Basic Facts About z-Scores

Vocabulary

Formulas For z-Scores

Quiz 4

Summary

# Bit More About z-Scores

The z-score indicates the distance an individual score is from the mean of a distribution. If a score is at the mean, it has a z-score of 0. Scores above the mean are positive and scores that are located below the mean are negative.

In practical terms, z-scores range from -3 to +3. A z of -3 indicates that the raw score is 3 standard deviations below the mean (at the extreme left end of the distribution). A z of 3 indicates that the raw score is at the extreme right end of the distribution.

Since z-scores are expressed in units of standard deviation, they are independent of the variable being measured. A z-score of -1.5 is one and a half standard deviations below the mean, regardless If z = .5, the score is located at one half standard deviation above the mean.

Composed of two parts, the z-score has both magnitude and sign. The magnitude can be interpreted as the number of standard deviations the raw score is away from the mean. The sign indicates whether the score is above the mean (+) or below the mean (-). To calculate the z-score, subtract the mean from the raw score and divide that answer by the standard deviation of the distribution. In formal terms, the formula is

Using this formula, we can find z for any raw score, assuming we know the mean and standard deviation of the distribution. What is the z-score for a raw score of 110, a mean of 100 and a standard deviation of 10? First, we find the difference between the score and the mean, which in this case would be 110-100 = 10. The result is divided by the standard deviation (10 divided by 10 = 1). With a z score of 1, we know that the raw score of 110 is one standard deviation above the mean for this distribution being studied.

Z-scores can be used to find an individual, standardize a distribution or set a cutoff. A z-score indicates a score’s distance from a mean, expressed in standard deviations. If a score is at the mean, z = 0. One standard deviation above the mean is indicated by z = 1. And one standard deviation below the mean is expressed as z = -1.

NOW YOU CHOOSE:

Day 4: z-Score

A Bit More About z-Scores

Even More About z-Scores

How To Calculate z-Scores

Practice Problems

Basic Facts About z-Scores

Vocabulary

Formulas For z-Scores

Quiz 4

Summary

# Basic Facts About Dispersion

October 22, 2008 by kltangen

Filed under Basic Facts

5 Measures of Dispersion

Range

Mean Absolute Deviation (Mean Variance)

Sum of Squares

Variance

Standard Deviation

NOW YOU CHOOSE:

Day 3: Dispersion

A Bit More About Dispersion

Even More About Dispersion

Range

MAD

Sum of Squares

Variance

Standard Deviation

How To Calculate

Range

MAD

Sum of Squares

Variance

Standard Deviation

Formulas For Dispersion

Practice Problems

More Practice Problems

Basic Facts About Dispersion

Vocabulary

Quiz 3

Summary

# Bit More About Dispersion

All measures of dispersion get larger when the distribution of scores is more widely varied. A narrow distribution (a lot of similar scores) has a small amount of dispersion. A wide distribution (lots of different scores) has a wide distribution. The more dispersion, the more heterogeneous (dissimilar) the scores will be.

Range is easy to calculate. It is the highest score minus the lowest score. If the highest score is 11 and the lowest score is 3, the range equals 8.

As the name suggests, mean variance (or mean absolute deviation) is a measure of variation from the mean. It is the average of the absolute values of the deviations from the mean. That is, the mean is subtracted from each raw score and the resulting deviations (called “little d’s”) are averaged (ignoring whether they are positive or negative).

Conceptually, Sum of Squares (abbreviated SS) is an extension of mean variance. Instead of taking the absolute values of the deviations, we square the critters (deviations), and add them up.

Variance of a population is always SS divided by N. This is true whether it is a large population or a small one. Variance of a large sample (N is larger than 30) is also calculated by Sum of Squares divided by N. If there are 40 or 400 in the sample, variance is SS divided by N.

However, if a sample is less than 30, it is easy to underestimate the variance of the population. Consequently, it is common practice to adjust the formula for a small sample variance. If N<30, variance is SS divided by N-1. Using N-1 instead of N results is a slightly larger estimate of variance and mitigates against the problem of using a small sample.

This measure of dispersion is calculated by taking the square-root of variance. Regardless of whether you used N or N-1 to calculate variance, standard deviation is the square-root of variance. If variance is 7.22, the standard deviation is 2.69. If variance is 8.67, the standard deviation equals 2.94.

Technically, the square-root of a population variance is called sigma and the square-root of a sample variance is called the standard deviation. As a general rule, population measures use Greek symbols and sample parameters use English letters.

NOW YOU CHOOSE:

Day 3: Dispersion

A Bit More About Dispersion

Even More About Dispersion

Range

MAD

Sum of Squares

Variance

Standard Deviation

How To Calculate

Range

MAD

Sum of Squares

Variance

Standard Deviation

Formulas For Dispersion

Practice Problems

More Practice Problems

Basic Facts About Dispersion

Vocabulary

Quiz 3

Summary

# Even More About Range

Like all measures of dispersion, the range of scores gets larger when the distribution of scores is more heterogeneous (dissimilar). The more homogeneous (similar) the scores, the smaller the range.

Range is easy to calculate. It is the highest score minus the lowest score. If the highest score is 11 and the lowest score is 3, the range equals 8. Range is the highest score minus the lowest score.

Naturally, if there is rounding involved, the range is from 11.49999 down to 2.50000. That is you would accept 11.4999 as still being 11 but 11.5000 would be considered a 12. And you would accept 2.5000 as being 3 but anything below that (2.4999 and lower) would be considered a 2. So some textbooks recommend adding 1 point to the range. According to this view, then, the range would be 11 minus 3 plus one; the range would equal 9.

For the purposes of our discussion, however, lets do it the easy way. Range is the high score minus the low score.

Although easy to calculate, range is not terribly helpful for describing the distribution. Without knowing what is being measured, a range of 12 is ambiguous. If we were measuring the number of points each basketball made during a game, a range of 12 would not be surprising. But if we were measuring the number of goals each hockey player made during a game, a range of 12 would be very unusual.

Range is a good way to check for input errors. If your were inputting scores from a 10-point quiz, a range of 72 would alert you to an input error. The maximum possible in a 10-point quiz is 10 and the lowest possible score is 0, the range should not be more than 10.

NOW YOU CHOOSE:

Day 3: Dispersion

A Bit More About Dispersion

Even More About Dispersion

Range

MAD

Sum of Squares

Variance

Standard Deviation

How To Calculate

Range

MAD

Sum of Squares

Variance

Standard Deviation

Formulas For Dispersion

Practice Problems

More Practice Problems

Basic Facts About Dispersion

Vocabulary

Quiz 3

Summary

# Even More About MAD

Although MAD sounds like anger or insanity, it’s not really the craziest statistic around. The real name is Mean Absolute Deviation. Or MAD, for short.

s the name suggests, mean absolute deviation (sometimes called mean variance) is a measure of deviation (variation) from the mean. To find the average amount of variation, the mean is subtracted from each score.

In the first column is a variable we’ll call X. The mean of this variable is 5. So 5 (column 2) is deducted from each score and the result forms column 3. Since the result is a measure of deviation from the mean, the third column is labeled d (little d).

7 5 2

5 5 0

5 5 0

5 5 0

3 5 -2

Mean variance sounds like it should be the mean of those little d’s (column 3). We would simply sum the column and divide by the number of scores. But there is a problem. When the little d’s are added up, they total zero (2+0+0+0-2=0).

But this is to be expected. We started at the mean, which is the balance point of the variable, and measured deviations from it. Since the mean is the center point of the distribution, deviations from it will always add up to 0. So we have two choices. We can take the absolute value of the deviations (which leads us to mean variance) or we can square them (as we’ll do in Sums of Squares below).

If we take the absolute value of the deviations, we ignore the sign (positive or negative) of each number. By ignoring the sign, the magnitude of the deviation is added and the result is no longer 0. In the above example, ignoring the positive and negative signs results in a sum of 4 (2+0+0+0+2) and a mean variance (average of the little d’s) of .80 (4 divided by 5).

So mean variance is the average of the absolute values of the deviations from the mean. The mean is subtracts from each raw score and the resulting little d’s are averaged (ignoring whether they are positive or negative).

The mean variance is a bit more complicated to calculate than range but more useful as a measure of dispersion. Mean variance is tied to the mean, gives a quick way to describe dispersion from the mean, and is useful when describing skewed distributions.

The down side is that mean variance doesn’t describe the underlying distribution. A mean variance of 7 is larger than a mean variance of 1.2, but it doesn’t describe the interrelationship of the scores well.

Day 3: Dispersion

A Bit More About Dispersion

Even More About Dispersion

Range

MAD

Sum of Squares

Variance

Standard Deviation

How To Calculate

Range

MAD

Sum of Squares

Variance

Standard Deviation

Formulas For Dispersion

Practice Problems

More Practice Problems

Basic Facts About Dispersion

Vocabulary

Quiz 3

Summary

# Even More About Sum of Squares (SS)

Like all measures of dispersion, Sum of Squares (SS) gets larger when the distribution of scores is more dissimilar (heterogeneous). The more homogeneous (similar) the scores, the smaller the distribution.

Deviation Method

Conceptually, Sum of Squares is an extension of mean variance. Instead of taking the absolute values of the deviations, we square the critters. For example:

X mean d d^{2
} 7 5 2 4

5 5 0 0

5 5 0 0

5 5 0 0

3 5 -2 4

The sum of the squared deviations is called Sum of Squares. In this example, the sum of the squared deviations is 8. The Sum of Squares (SS for short) is the sum of the squared deviations. Like all measures of dispersion, the larger the number, the more dispersed the distribution of raw scores. The smaller the SS, the less dispersed the scores are.

This “deviation method” of calculating Sum of Squares is illustrate that it is a measure of dispersion from the mean. After using this method several times, it should be clear that the Sum of Squares is the sum of the squared deviations. It is a measure of squared deviations from the mean.

Once this concept is clear, you’ll be ready to know the secret: there is an easier way to calculate Sum of Squares.

Raw Score Method

The problem with the deviation method is clearest when the mean is not an integer. When the mean is 5, it’s not hard to subtract it from every score. When the mean 5.387, it is difficult to know how many places to carry out each of the sub-answers. It’s not impossible to do; it’s just a pain.

It seems like some mathematician with nothing better to do must have come up with a easier way to calculate Sum of Squares. And, in truth, there is an easier way.

The raw score method only uses the raw scores; there are no deviations to calculate. Here is the formula we use:

Here’s how the process works. Assume this is the distribution at issue:

X

11

7

3

4

5

8

First, each number is squared:

X X^{2
} 11 121

7 49

3 9

4 16

5 25

8 64

Now, we sum each column. The sum of the first column is 38 (it’s called the Sum of X). And the sum of the second column is 284; this number is called the Sum of the X-squareds.

Next, we square the sum of X (38 times itself = 1444) and divide it by N. Since N = 6, we divide 1444 by 6 (which equals 240.67).

Fourth, we recall the sum of the X^{2} and subtract 240.67 from it. So 284 minus 240.67 = 43.33. The Sums of Squares is 43.33.

Both Ways At Once

Let’s compare the two methods. They will produce the same results but the raw score method is much easier to calculate.

X X^{2} d d^{2
} 11 121 4.67 21.78

7 49 .67 .44

3 9 -3.33 11.11

4 16 -2.33 6.44

5 25 -1.33 1.78

Sum 38 284 0 43.33

N = 6

Mean = 6.66

Sum of Squares 44.33 43.33

Day 3: Dispersion

A Bit More About Dispersion

Even More About Dispersion

Range

MAD

Sum of Squares

Variance

Standard Deviation

How To Calculate

Range

MAD

Sum of Squares

Variance

Standard Deviation

Formulas For Dispersion

Practice Problems

More Practice Problems

Basic Facts About Dispersion

Vocabulary

Quiz 3

Summary