We now understand how to describe and present our data visually and numerically. These simple tools, and the principles behind them, will help you interpret information presented to you and understand the basics of a variable. Moving forward, we now turn our attention to how scores within a distribution are related to one another, how to precisely describe a scores location within the distribution, and how to compare scores from different distributions.
In many situations, it is useful to have a way to describe the location of an individual score within its distribution. One approach is the percentile rank. The percentile rank of a score is the percentage of scores in the distribution that are lower than that score. Percentiles are useful for comparing values.
Consider, for example, the distribution of Rosenberg Self-esteem scores we used in chapter 2. For any score in the distribution, we can find its percentile rank by counting the number of scores in the distribution that are lower than that score and converting that number to a percentage of the total number of scores.Self-Esteem ScoresFrequencyCumulative FrequencyCumulative Percentage243401002353792.52210328021822552051435193922.518361517037.516237.515112.5
Table 1. Frequency table for Rosenburg self-esteem scores
Notice, for example, that five of the students represented by the data in the table had self-esteem scores of 23. In this distribution, 32 of the 40 scores (80%) are lower than 23 (note that you can see for score 22 showing cumulative frequency as 32). Thus, for students with a score of 23, they have a percentile rank of 80 percent. It can also be said that they scored at the 80th percentile. Remember that percentile rank by counting the number of scores in the distribution that are lower than that score and converting that number to a percentage of the total number of scores (32/40 = 80%). Percentile ranks are often used to report the results of standardized tests of ability or achievement. If your percentile rank on a test of verbal ability were 40, for example, this would mean that you scored higher than 40% of the people who took the test.
The normal distribution is the most important and most widely used distribution in statistics. It is sometimes called the bell curve, although the tonal qualities of such a bell would be less than pleasing. It is also called the Gaussian curve of Gaussian distribution after the mathematician Carl Friedrich Gauss. Lets review a little bit about the normal distribution. The normal distribution is described in terms of two parameters: the mean (which you can think of as the location of the peak), and the standard distribution (which specifies the width of the distribution). The bell-like shape of the distribution never changes, only its location and width. The normal distribution is commonly observed in data collected in the real world, as we have already seen in Chapter 3 and in chapter 7 we will learn more about why this occurs.
Photo of Gauss Monument dedicated to the mathematician, geodesist and astronomerCarl Friedrich Gauß. It is placed in his place of birth Brunswick. Photo credit
Strictly speaking, it is not correct to talk about the normal distribution since there are many normal distributions. Normal distributions can differ in their means and in their standard deviations. Figure 1 shows three normal distributions. The green (left-most) distribution has a mean of -3 and a standard deviation of 0.5, the distribution in red (the middle distribution) has a mean of 0 and a standard deviation of 1, and the distribution in black (right-most) has a mean of 2 and a standard deviation of 3. These as well as all other normal distributions are symmetric with relatively more values at the center of the distribution and relatively few in the tails. What is consistent about all normal distribution is the shape and the proportion of scores within a given distance along the x-axis. We will focus on the Standard Normal Distribution (also known as the Unit Normal Distribution), which has a mean of 0 and a standard deviation of 1 (i.e. the red distribution in Figure 1).
Figure 1. Normal distributions differing in mean and standard deviation.
Seven features of normal distributions are listed below.
These properties enable us to use the normal distribution to understand how scores relate to one another within and across a distribution. But first, we need to learn how to calculate the standardized score than make up a standard normal distribution.
As we learned in earlier lessons, population mean (µ) and population standard deviation (σ) are methods for describing an entire distribution of scores using individual scores. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading. A z-score is a standardized version of a raw score (x) that gives information about the relative location of that score within its distribution. Z-scores are standardized scores that identify and describe the exact location of every score within a distribution. By transforming our values (raw score) we can compare z-scores across different samples or groups and make meaningful comparisons. Each value in the distribution has a z-score that can be calculated to standardize for comparison.
Lets say that you received a score of 76 on your Chemistry exam and your friend receives a score of 76 on her Physics exam. Who is doing better in class? It is hard to say because we do not have enough information. This is an example of how z-scores can facilitate meaningful comparisons.The z score for a particular individual is the difference between that individuals score and the mean of the distribution, divided by the standard deviation of the distribution.
Formulas to calculate Z scoresPopulationSample
Note that it is essentially the same formula where the appropriate symbols for mean and standard deviation have been used depending on if working with population or sample data.
As you can see, z-scores combine information about where the distribution is located (the mean/center) with how wide the distribution is (the standard deviation/spread) to interpret a raw score (x). Specifically, z-scores will tell us how far the score is away from the mean in units of standard deviations and in what direction. Z-scores transforms raw scores into units of standard deviation above or below the mean. This transformation provides a reference using the standard normal distribution. If we are given a Z score we know where, relative to the mean, the Z score and raw score lies.
The value of a z-score has two parts: the sign (positive or negative) and the magnitude (the actual number). The sign of the z-score tells you in which half of the distribution the z-score falls: a positive sign (or no sign) indicates that the score is above the mean and on the right hand-side or upper end of the distribution, and a negative sign tells you the score is below the mean and on the left-hand side or lower end of the distribution. The magnitude of the number tells you, in units of standard deviations, how far away the score is from the center or mean. The magnitude can take on any value between negative and positive infinity, but for reasons we will see soon, they generally fall between -3 and 3.
Z-scores & the standard normal distribution
Lets look at some examples. A z-score value of -1.0 tells us that this z-score is 1 standard deviation (because of the magnitude 1.0) below (because of the negative sign) the mean.
Figure 2. z-score of -1
Similarly, a z-score value of 1.0 tells us that this z-score is 1 standard deviation above the mean. Thus, these two scores are the same distance away from the mean but in opposite directions. A z-score of -2.5 is two-and-a-half standard deviations below the mean and is therefore farther from the center than both of the previous scores, and a z-score of 0.25 is closer than all of the ones before.
We can convert raw scores into z-scores to get a better idea of where in the distribution those scores fall. Lets say we get a score of 68 on an exam (X=68). We may be disappointed to have scored so low, but perhaps it was just a very hard exam. Having information about the distribution of all scores in the class would be helpful to put some perspective on ours. We find out that the class got an average score (M) of 54 with a standard deviation (s) of 8. To find out our relative location within this distribution, we simply convert our test score into a z-score.
z =(𝑋 M)/s = (68 54)/8= 1.75
We find that we are 1.75 standard deviations above the average, above our rough cut off for close and far. Suddenly our 68 is looking pretty good!
Figure 3. Raw and standardized versions of a single score
Figure 3 shows both the raw score and the z-score on their respective distributions. Notice that the red line indicating where each score lies is in the same relative spot for both. This is because transforming a raw score into a z-score does not change its relative location, it only makes it easier to know precisely where it is.
Lets go back to our Chemistry and Physics exam score comparisons. Each student received a score of x =76 on the exam. Assume that:
A z score indicates how far above or below the mean a raw score is, but it expresses this in terms of the standard deviation. The z-scores for our example are above the mean.
When we compare the position of the test score X = 76 it is clear that these two distributions are very different and that the Chemistry score has a higher position in the distribution.
As mentioned earlier, z-scores are also useful for comparing scores from different distributions. Lets say we take the SAT and score 501 on both the math and critical reading sections. Does that mean we did equally well on both? Scores on the math portion are distributed normally with a mean of 511 and standard deviation of 120, so our z- score on the math section is zmath= 501 511/120= 0.08 which is just slightly below average (note that use of math as a subscript; subscripts are used when presenting multiple versions of the same statistic in order to know which one is which and have no bearing on the actual calculation). The critical reading section has a mean of 495 and standard deviation of 116, so zcreading= 501 495/116= 0.05. So even though we were almost exactly average on both tests, we did a little bit better on the critical reading portion relative to other people.
Finally, z-scores are incredibly useful if we need to combine information from different measures that are on different scales. Lets say we give a set of employees a series of tests on things like job knowledge, personality, and leadership. We may want to combine these into a single score we can use to rate employees for development or promotion, but look what happens when we take the average of raw scores from different scales, as shown in Table 2:
Table 2. Raw test scores on different scales (ranges in parentheses).
Because the job knowledge scores were so big and the scores were so similar, they overpowered the other scores and removed almost all variability in the average. However, if we standardize these scores into z-scores, our averages retain more variability and it is easier to assess differences between employees, as shown in Table 3.
Table 3. Standardized scores
Setting the scale of a distribution
Another convenient characteristic of z-scores is that they can be converted into any scale that we would like. Here, the term scale means how far apart the scores are (their spread) and where they are located (their central tendency). In other words, we can convert that value into its original raw score (X) if the mean and standard deviation are known. We can still do this using the Z score formula we have been using so far this lesson. We just need to rearrange the variables so we are solving for X instead of Z.
The formulas for transforming z to x are:
Note: these are just simple rearrangements of the original formulas for calculating z from raw scores.
A problem is that these new z-scores arent exactly intuitive for many people. We can give people information about their relative location in the distribution (for instance, the first person scored well above average). Another route we can do is to take the z-scores and transform them to a known distribution, like the traditional IQ distribution.
Lets say we have z-scores of 1.71, .43, and .80 after converting their raw intelligence test score to a z-score. We can translate these z-scores into the more familiar metric of IQ scores, which have a mean of 100 and standard deviation of 16. We can use the transforming formula from above: X = z * SD + M
X = 1.71 16 + 100 = 127.36, so IQ score of 127
X = 0.43 16 + 100 = 106.88, so IQ score of 107
X = 0.80 16 + 100 = 87.20, so IQ score of 100
We rounded the values to 127, 107, and 87, respectively, for convenience.
Z-scores and the Area under the Curve
Even though we can use a z-score as a measure of relative standing for any shape of frequency distribution, we commonly use z-scores in this class when discussing normal distributions. They provide a way of describing where an individuals score is located within a distribution and are sometimes used to report the results of standardized tests.
Z-scores and the standard normal distribution go hand-in-hand. A z-score will tell you exactly where in the standard normal distribution a value is located, and any normal distribution can be converted into a standard normal distribution by converting all of the scores in the distribution into z-scores, a process known as standardization. We will also see that one can identify the percentile for each z-score in a normal distribution.
Since a z-score tells us how far above or below the mean a particular raw score lies (in standard deviation units), we can use z-scores in conjunction with the empirical rule. We can use z-scores to simplify the earlier statements we made regarding the Empirical Rule (68-95-99 rule):
Take a minute to examine Figure 4 to identify these areas. For example, you can see adding up the 2 areas between z = -1 to z = 1, you get 68.2%. Because z-scores are in units of standard deviations, this means that 68% of scores fall between z = -1.0 and z = 1.0 and so on. We call this 68% (or any percentage we have based on our z-scores) the proportion of the area under the curve. Remember, these percentages remain true only if our sample or population is normally distributed!
Figure 4. Z-score indicating percentiles in a standardized normal distribution.
Any area under the curve is bounded by (defined by, delineated by, etc.) by a single z-score or pair of z-scores. An important property to point out here is that, by virtue of the fact that the total area under the curve of a distribution is always equal to 1.0 (see section on Normal Distributions at the beginning of this chapter), these areas under the curve can be added together or subtracted from 1 to find the proportion in other areas. For example, we know that the area between z = -1.0 and z = 1.0 (i.e. within one standard deviation of the mean) contains 68% of the area under the curve, which can be represented in decimal form at 0.6800 (to change a percentage to a decimal, simply move the decimal point 2 places to the left). Because the total area under the curve is equal to 1.0, that means that the proportion of the area outside z= -1.0 and z = 1.0 is equal to 1.0 0.6800 = 0.3200 or 32% (see Figure 5 below). This area is called the area in the tails of the distribution. Because this area is split between two tails and because the normal distribution is symmetrical, each tail has exactly one-half, or 16%, of the area under the curve.
Figure 5. Shaded areas represent the area under the curve in the tails
Additionally, z-scores provide one way of defining outliers. For example, outliers are sometimes defined as scores that have z scores less than 3.00 or greater than +3.00. In other words, they are defined as scores that are more than three standard deviations from the mean. Some researchers will define outliers as greater than 2 standard deviations from the mean.
We will have much more to say about percentiles in a distribution in the coming chapters. As it turns out, this is a quite powerful idea that enables us to make statements about how likely an outcome is and what that means for research questions we would like to answer and hypotheses we would like to test. But first, we need to make a brief foray into some ideas about probability.
Having read this chapter, you should be able to:
Exercises Chapter 6
Answers to Odd-Numbered Exercises Ch. 6
3. X = 4.2, s = 1.64; z = -1.34, -0.73, 0.49, 0.49, 1.10
5. 1. 2 standard deviations below the mean, far
2. 1.25 standard deviations above the mean, near
3. 3.5 standard deviations above the mean, far
4. 0.34 standard deviations below the mean, near
7. z = 0.75, 0.56, -2.75, -0.75, 2.19, -0.06
9. 1. -0.50, 2. 0.25, 3. 3.00, 4.1.10
11. Z = (52 40)/7 = 1.71
Z = (43 40)/7 = 0.43
Z = (34-40)/7 = -0.80