Psychology
Z-Score
A Z-score in psychology is a statistical measure that indicates how many standard deviations a data point is from the mean of a group of data. It is used to compare and interpret individual scores within a distribution, allowing psychologists to understand the relative position of a particular score and make meaningful comparisons across different measures and populations.
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11 Key excerpts on "Z-Score"
eBook - PDF- Sherri Jackson(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Conversion to a z score is a statistical technique that is appropriate for use with data measured on a ratio or interval scale of measurement (scales for which means are calculated). Let’s use the formula to calculate the z scores for the previously mentioned student’s psychology and English exam scores. The necessary information is summarized in Table 6.1. To calculate the z score for the English test, we first calculate the dif-ference between the score and the mean and then divide by the standard deviation. We use the same process to calculate the z score for the psychol-ogy exam. These calculations are as follows: z English 5 X 2 X S 5 91 2 85 9.58 5 6 9.58 5 1 0.626 z Psychology 5 X 2 X S 5 86 2 74 13.64 5 12 13.64 5 1 0.880 z score (standard score) A number that indicates how many standard deviation units a raw score is from the mean of a distribution. TABLE 6.1 Raw score ( X ), sample mean ( X ), and standard deviation ( S ) for English and psychology exams X ( X ) S English 91 85 9.58 Psychology 86 74 13.64 Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 84 ■ ■ CHAPTER 3: MODULE 6 The individual’s z score for the English test is 0.626 standard deviations above the mean, and the z score for the psychology test is 0.880 standard deviations above the mean.
No longer available |Learn more- K. Paul Nesselroade, Jr., Laurence G. Grimm, K. Paul Nesselroade, Jr(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
This way of comparing scores from different scales of measurement is very useful in the social and behavioral sciences as well as in the field of education. We can ask, for instance, if a person is more depressed than anxious, more par-anoid than manic, or better at math than at reading. Although the scales of the tests are designed to tap different traits and abilities, and each scale has its own mean and standard deviation, by standardizing the raw scores an examiner can easily make cross-scale comparisons. 138 5 The Normal Curve and Transformations ( z = +1). What z score would be assigned to a score of 16? Since 16 is one stand-ard deviation below the mean (20 -4), a score of 16 would transform to a z score of -1. A z score is also called a standard unit or standard score . When all of the raw scores from a normal distribution have been transformed into z score s , the resulting distribution is called the standard normal distribution . The standard normal distribution is a special distribution; it has a mean of 0 and a standard deviation of 1. This point is so important – it bears repeating; any normal dis-tribution of raw scores if converted into z scores, no matter the mean or the standard deviation, will take the shape of the standard normal distribution, hav-ing a mean of 0 and a standard deviation of 1. This makes the standard normal distribution very special. Two z score formulas are provided; one is used to transform the scores of a population, while the other is used to transform the scores of a sample.
eBook - ePub- Marion Marshall, Marion E. Marshall(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
Figure 3.2 and note that 50 + 34 = 84%. To use another example, a Z-Score of –1.0 would translate to the 16th percentile rank (50 – 34 = 16).More than likely, the test maker will not use these Z-Scores but will have converted them to “standard scores” with a different scale, like the SAT does. Given the earlier SAT example, the 600 score would be generated from a +1.0 Z-Score. Suffice it to say that this can seem confusing if every test uses a different scale though many use a scale similar to cognitive (IQ) tests, which have a mean of 100 and an SD of 15. It is interesting to note that “Standard” scores are not really standard. The reason is the word, as it is used here, refers to “standardizing” the scores, not that they all use the same scale. In fact, the definition of “standard score” is what is called the Z-Score here. This is why using percentile ranks can help you with comparing results across tests and most readers are used to thinking about and comparing results using percentile ranks. While the “standard score” scales may vary, the percentiles can be compared directly. Just be clear when explaining the assessment results to parents to clarify the difference between percentile rank and percentage correct . See Chapter 6 to learn how to explain tests results to parents and avoid this potential misunderstanding, and see Wright and Darr Wright (2016) for a detailed, carefully constructed and “parent friendly” explanation of the psychometrics of assessment scores is on: www.wrightslaw.com/advoc/articles/tests_measurements.htmlRemember that, while you need to know how the scores are derived, you will not be required to calculate them. The scores will be accessed in a table in the examiner’s manual or, if the test scoring is computer-based, from the data file in the computer. They will be accurate if you are using a well-constructed and technically reliable instrument, have computed the chronological age correctly, and have calculated the raw scores (or composite scores based on the raw scores) accurately. What matters is that you understand what the scores mean.
eBook - PDF- Frederick Gravetter, Larry Wallnau(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
The z -score distribution will have the same shape as the distribution of raw scores, and it always will have a mean of 0 and a standard deviation of 1. 5. When comparing raw scores from different distribu-tions, it is necessary to standardize the distributions with a z -score transformation. The distributions will then be comparable because they will have the same parameters ( μ = 0, σ = 1). In practice, it is neces-sary to transform only those raw scores that are being compared. SUMMARY 153 Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 6. In certain situations, such as psychological testing, a distribution may be standardized by converting the original X values into z -scores and then converting the z -scores into a new distribution of scores with predetermined values for the mean and the standard deviation. 7. In inferential statistics, z -scores provide an objective method for determining how well a specific score rep-resents its population. A z -score near 0 indicates that the score is close to the population mean and therefore is representative. A z -score beyond + 2.00 (or − 2.00) indicates that the score is extreme and is noticeably different from the other scores in the distribution. raw score (134) z -score (135) deviation score (136) z -score transformation (141) standardized distribution (142) standardized score (145) KEY TERMS General instructions for using SPSS are presented in Appendix D. Following are detailed instructions for using SPSS to Transform X Values into z -Scores for a Sample.
- Robert Warner(Author)
- 2015(Publication Date)
- Elsevier(Publisher)
P value, whether the data point is larger or smaller than the mean of the population to which the data point is being compared.Z scores are broadly analogous to the well-known T test in statistics. Both Z scores and the T test are used to test the null hypothesis, i.e., to determine the likelihood that there is no meaningful difference from the mean of a comparison population. Whereas a Z score tests the null hypothesis concerning the difference between each data point and the mean of a comparison population, the T test tests the null hypothesis concerning the means of two different populations of data.It is important to be able to test the null hypothesis for individual data points. This is because the individual values in most sets of data can be expected to vary to some extent. Using Z scores permits one to determine for each data point that differs from a population’s mean, whether it is likely to represent a true abnormality or to merely be an example of the expected random variation among the values in a population of data.One way to regard the Z score is as a tool for solving the classical problem of distinguishing between signal and noise. In the context of data analysis, noise consists of chance variations among the values of data points that fail to reveal meaningful information. In the present context, statistically significant Z scores indicate which members of a set of data are most likely to represent an actual signal, i.e., to convey meaningful information. Conversely, statistically insignificant Z scores are more to indicate data points that represent random noise. The determination of what P value should be considered to indicate a statistically significant difference depends upon various factors. One factor is whether it is considered more important not to miss a given abnormality (such as a very serious medical condition) or more important not to incorrectly identify apparent abnormalities whose presence would be inconsequential. Another factor is whether one is making comparisons that involve only one versus several different parameters. Comparisons involving multiple parameters require more stringent P values and therefore higher absolute values of Z scores. If the context in which the comparison of the data requires greater stringency, then lower P
- Bruce M. King, Patrick J. Rosopa, Edward W. Minium(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
For example, in a distribution that has = 100 and = 20, a score of 120 may be expressed as a z score of +1, indicating that the score is 1 standard deviation above the mean of the distribution. Similarly, a score of 60 in the same distribution is expressed as a z score of −2.0, because it is 2 standard deviations below the mean. The formula for a z score is DEFINITIONAL FORMULA FOR z SCORE IN A POPULATION z = X − X X (5.5a) DEFINITIONAL FORMULA FOR z SCORE IN A SAMPLE z = X − X S X (5.5b) z = X − X X formula for a z score in a population z = X − X S X formula for a z score in a sample The z score makes it possible, under some circumstances, to compare scores that originally had different units of measurement because after conversion each has a common unit of measurement. For example, how do we compare the performance on a history and a chemistry exam? That is like trying to compare apples to oranges. Suppose that Helen, a fifth grader, earned an 80 on her American history exam and a 65 on her arithmetic exam in the same class. Is she doing better in history or arithmetic? First, we need to know how the rest of Helen’s class did on the two exams. Suppose that the mean score on the history exam was 65 and that the mean score on the arithmetic exam was 50. Helen thus scored 15 points higher than the mean for the class on both exams. Can we conclude that she performed equally well in both subjects? We still do not know because we don’t know whether 15 points on the history exam is the same as 15 points on the arithmetic exam. Suppose also that the standard deviation on Helen’s history exam was 15, but that the standard deviation on her arithmetic exam was only 7.5. Now we can determine in which subject she did better. Helen’s history score of 80 has a z value of +1.0, because it was +1.0 standard deviations greater than the mean for her class [(80 − 65)∕15].
eBook - PDFStatistics for the Social Sciences
A General Linear Model Approach
- Russell T. Warne(Author)
- 2017(Publication Date)
- Cambridge University Press(Publisher)
The i subscript in z i indicates that each individual score in a dataset has its own Z-Score. There are two steps in calculating Z-Scores. The first step is in the numerator where the sample mean is subtracted from each score. (You may recognize this as the deviation score, which was discussed in Chapter 4.) This step moves the mean of the data to 0. The second step is to divide by the standard deviation of the data. Dividing by the standard deviation changes the scale of the data until the deviation is precisely equal to 1. Guided Practice 5.1 shows how to calculate Z-Scores for real data. The example Z-Score calculation in Guided Practice 5.1 illustrates several principles of Z-Scores. First, notice how every individual whose original score was below the original mean (i.e., X = 25.69) has a negative Z-Score, and every individual whose original score was above the original mean has a positive Z-Score. Additionally, when comparing the original scores to the Z-Scores, it is apparent that the subjects whose scores are closest to the original mean also have Z-Scores closest to 0 – which is the mean of the Z-Scores. Another principle is that the unit of Z-Scores is the standard deviation, meaning that the difference between each whole number is one standard deviation. Finally, in this example the person with the lowest score in the original data also has the lowest Z-Score – and the person with the highest score in the original data also has the highest Z-Score. In fact, the rank order of subjects’ scores is the same for both the original data and the set of Z-Scores because linear transformations (including converting raw scores to Z-Scores) do not change the shape of data or the ranking of individuals’ scores. These principles also apply to every dataset that is converted to Z-Scores. There are two benefits of Z-Scores. The first is that they permit comparisons of scores across different scales.
- Martin Lee Abbott(Author)
- 2016(Publication Date)
- Wiley(Publisher)
There has to be room under the curve for these kinds of possibilities. • The inflection point of the standard normal curve is at the point of the (negative and positive) first SD unit. This point is where the steep decline of the curve slows down and widens out. (This is a helpful visual cue to an advanced pro- cedure called factor analysis, which uses a scree plot to help decide how many factors to use from the results.) THE STANDARD NORMAL SCORE: Z SCORE When we refer to the standard normal deviation, we speak of the “z score,” which is a very important measure in statistics. A z score is a raw score expressed in SD units. Thus, a z score of 0.67 would represent a raw score that is two-thirds of one SD to the right of the mean. These scores are shown on the x-axis of Figure 4.1 and represent the SD values that define the areas of the distribution. Thus, +1 SD unit to the right of the mean contains 34.13% of the area under the standard normal curve (from the mean), and we can refer to this point as a z score of +1. So, if a given AP test score had a z score of 3.5, we would recognize immediately that this school would have an inordinately high raw score, relative to the other values, since it would fall three and Mean Z = –1.96 Figure 4.2 The location of z = (−)1.96. 80 THE NORMAL DISTRIBUTION one-half SDs above the mean where there is only an extremely small percent of the curve represented. Because it has standardized meaning, the z score allows us to understand where each score resides compared to the entire set of scores in the distribution. It also allows us to compare one individual’s performance on two different sets of (normally distributed) scores. It is important to note that z scores are expressed not just in whole numbers but as decimal values, as I used in the example earlier. Thus, a z score of −1.96 would indicate that the raw score is slightly less than two SDs below the mean on a standard normal curve as shown in Figure 4.2.
eBook - PDFEducational Assessment
Tests and Measurements in the Age of Accountability
- Robert J. Wright(Author)
- 2007(Publication Date)
- SAGE Publications, Inc(Publisher)
If the score is equal to one standard deviation above the mean, the Z-Score will be +1.0. To find any z -score, it is necessary to know the original score, sample mean, and the sample’s standard deviation. The z -score can be found as follows: In this equation, the value of X is the individual score being changed into a z -score (standard score). As an example, if an applicant to graduate school has a GPA of 3.25 from a sample with a mean of X — = 2.50, and a standard deviation of 0.50, that student would have a z -score of +1.50. This indicates that this student had a college grade point average that was equiv-alent to one and a half standard deviation units above the mean. z = X − X s z = 3 : 25 − 2 : 5 0 : 5 z = 1 : 5 Chapter 3 The Measurement and Description of Variables – – 99 Percentiles The consistency of the normal curve makes it possible to interpolate z -scores into percentiles. For example, the score point in the center of the Gaussian normal curve ( z = 0.0) is the 50th percentile. That point has half of all scores below and above it. The Gaussian normal curve starts and ends at infinity. Because of that, there can never be a zero percentile point with these trans-formed scores. Likewise there can never be a score equal to the 100th percentile. Percentiles represent an ordinal transformation of data. The concept behind percentiles is that each percentile represents 1/100th of the data. The first percentile includes all those data points arranged from the lowest score to the point where 1% of the data are included. The first quartile is the 25th percentile, the point that cuts off the lowest 25% of the scores in the data set. The second quintile is the point that cuts off the lowest 40% of the scores from the data set. The median is the 50th percentile, and the seventh decile is a score that cuts off the lowest 70% of the data.
eBook - PDFPsychological Testing
Principles, Applications, and Issues
- Robert Kaplan, Dennis Saccuzzo(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
This system assumes that the scores are distributed normally. Now try an example that puts a few of these concepts together. Suppose you get a 60 on a social psychology examination. You learned in class that the mean for the test was 55.70 and that the standard deviation was 6.08. If your professor uses the grading system that was just described, what would your grade be? To solve this problem, first find your Z score. Recall the formula for a Z score: Z 5 X i 2 X S So your Z score would be Z 5 60 2 55.70 6.08 5 4.30 6.08 5 .707 Looking at Table 2.4, you see that .707 is greater than .25 (the cutoff for a B) but less than 1.04 (the cutoff for an A). Now find your exact standing in the class. To do this, look again at Appendix 1. Because the table gives Z scores only to the second decimal, round .707 to .71. You will find that 76.11% of the scores fall below a Z score of .71. This means that you would be in approximately the 76th percentile, or you would have performed better on this examination than approximately 76 out of every 100 students. McCall’s T There are many other systems by which one can transform raw scores to give them more intuitive meaning. One system was established in 1939 by W. A. McCall, who originally intended to develop a system to derive equal units on mental quantities. TABLE 2.4 Z Score Cutoffs for a Grading System Grade Percentiles Z score cutoff A 85–100 1.04 B 60–84 .25 C 20–59 2 .84 D 6–19 2 1.56 F 0–5 ,2 1.56 Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 CHAPTER 2 ● Norms and Basic Statistics for Testing 49 He suggested that a random sample of 12-year-olds be tested and that the distri-bution of their scores be obtained. Then percentile equivalents were to be assigned to each raw score, showing the percentile rank in the group for the people who had obtained that raw score.
- Martin Lee Abbott(Author)
- 2014(Publication Date)
- Wiley(Publisher)
z -score data and the descriptive information about the distribution. First, we should use the information we have to estimate what a solution might be.What can we observe generally? Certainly, because z scores are expressed in standard deviation units on the standard normal distribution, the student probably did not perform highly on the test (i.e., because negative scores are on the left of the distribution—in this case more than 1.5 SDs to the left of the mean). Consider the following formula, which is derived from the z -score formula listed above:In this formula, the z score we need to transform to a raw score (X ) is known, along with the SD and mean of the raw score distribution. Substituting the values we listed above, we obtainSo, we can inform the student’s mother that her child received 75.40 on the test. Although we don’t know how that might translate into a teacher’s grade, it probably has more meaning to a parent who does not normally see z scores. To the researcher, however, the z score contains more information.TRANSFORMING CUMULATIVE PROPORTIONS TO z SCORES
Another situation may arise in which we have cumulative proportions or percentiles available and wish to transform them to z scores. This is a fairly easy step because both are based on z scores.In our previous example, the student’s inquisitive mother would probably have been given a percentile rather than a z score because the educational system uses percentiles extensively as the means to make comparisons among scores. Here is a brief example, again using the mother. Suppose the student’s mother was told that her son received a score that was at the 60th percentile. What would be the z
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