PS115 Lecture 03 Handout Z- Scores & THE Normal Distribution PDF

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PS115 Lectures 3 & 4a – Weeks 4 & 5

Z-Scores and the Normal Distribution The Standard Normal Distribution – Standardized Scores and Z-Scores

We can ‘label’ a score by how many standard deviations above or below the mean the score is (‘above’ = higher than; ‘below’ = lower than). These are called standardized scores For normally distributed variables this standardized score is called the z-score

Example 1: Scores on a verbal reasoning test have a mean of 60 and a standard deviation of 10. Express the following test scores as standardised scores: (a) 80, (b) 45, (c) 52.

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Example 2: A variable is normally distributed with a mean of 6.8 and a standard deviation of 1.8 Express the following scores as z-scores: (a) 5.2, (b) 6.4, (c) 8.1.

If you know the z-score you can convert (“transform”) these onto a scale with a specified mean and standard deviation. Example 3: Convert the following z-scores onto a scale with a mean of 50 and a std. deviation of 10: (a) z = 3, (b) z = –2, (c) z = 1.5, (d) z = –0.3

Example 4: Transform z = 1.4 and z = –2.3 onto a scale with a mean = 20 and std. dev. = 4

Formulae for converting to and from z-scores (or standardized scores) To calculate the z-score for a score (𝑿) when the mean is µ and the std. deviation is σ :

𝒛!! = !!!

𝑿! − !𝝁! 𝝈

(µ µ represents the mean of a population, σ represents the standard deviation of a population) To calculate a score (𝑿) on a scale with a mean of µ and a standard deviation of σ when you know the z-score:

𝑿!! = !!!𝝁! + !!𝒛𝝈 -2-

Using Tables to Find the Proportion of Scores in a Specified Interval Remember we said that for a normally distributed variable: 68.3% of scores are within one standard deviation of the mean (i.e., between z = –1 and z = 1) 95.4% of scores are within two standard deviations of the mean (i.e., between z = –2 and z = 2)

The mathematical properties of the normal distribution allow us to specify the proportion of scores that should fall above or below any specified z-value. This corresponds to telling us what proportion of the area under a normal curve is to the right or to the left of that value. These “facts” can be found in tables. This is a “cut down” version from Howell (2014), showing this information for some values of z. A full set of tables is also provided (in the textbook, and as a handout). Proportion of scores falling above/below a specified z-score:

Remember: The area under a portion of the curve indicates the proportion of scores in that region.

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We can also specify the proportion of scores that fall between any two specified z-values. This corresponds to telling us what proportion of the area under a normal curve is in between these two values of z. Proportion of scores falling in between two specified z-scores:

Example 5: Use Tables for the Standard Normal Distribution to find out the proportion of scores in the standard normal distribution for which: (a)

z > 1.04 (“z is greater than 1.04”)

(b)

z > –2.02

(c)

z < –1.98 (“z is less than –1.98”) {“less than a negative score” = “a lower score”, “a more negative score”}

(d)

–1.96 < z < 1.96 (“z is greater than –1.96 but less than 1.96”)

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Example 6(a) In a reading test, Jack obtained a score of z = –2.05 and Katie obtained a score of z = – 1.05. Given that the scores on this test are normally distributed for pupils of the same age as Jack and Katie, what proportion of pupils taking the test will have a score that falls in between Jack’s Score and Katie’s score?

Example 6(b) In a test of reasoning ability for which scores are normally distributed, Anna scored z = –1.45 and Bill scored z = +1.45. What percentage of the people taking this test are expected to obtain a score that is higher than Bill’s score OR lower than Anna’s score.

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Sometimes you will have to convert to z-scores first... Example 7(a): For a normally distributed variable with a mean of 300 and a standard deviation of 50, what proportion of scores are greater than 251.4? (This answer is also the probability of randomly sampling a score greater than 251.4.)

Example 7(b): Scores on a particular test of cognitive ability are normally distributed with a mean of 400 and a standard deviation of 100. What percentage of scores are greater than 505 or less than 295?

To extremity and beyond … a short note on terminology Sometimes people describe scores as being “beyond” a particular z-score. This means that they are referring to scores that are in the tails of the distribution (i.e., scores that are above a positive z-score, or scores that are below a negative z-score). For instance, in Example 7(b), we could have asked: “What percentage of scores are beyond 505 or beyond 295?” – and this would have meant the same thing as the question that was written above The term “extreme” is often used in a similar way. In Example 7(b), we could have said: “What percentage of scores are more extreme than 505 or than 295?” – and again this would have had the same meaning as what was written above.

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PS115 Lecture 4a – Week 5

Finding a Z-Score for a Known Percentage or Proportion of Scores This uses the Tables for the Normal Distribution to “reverse” the process of finding a proportion from a given z-score: Lecture 3: Lecture 4a:

you know a z-score you know a proportion

→ →

find a proportion find the appropriate z-score

Example 8: Using Tables for the Normal Distribution: (a) Find the positive z-score for which the smaller portion is 0.1635

(b) Find the negative z-score for which the larger portion is 0.68

(c) Find the negative z-score for which the mean-to-z portion is 0.43

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Example 9(a): 85% of scores in the standard normal distribution are above what z-score?

Example 9(b): For a normally distributed variable with a mean of 10 and a standard deviation of 2, the lowest 2.5% of scores are below what value on the scale?

Example 10(a): Between which two z-scores are the 86% of scores that lie closest to the mean?

Example 10(b): For a normally distributed population with a mean of 100 and a standard deviation of 50, find the two cut-off scores that define the most extreme 15% of scores?

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Reading for Lectures 3 and 4a Howell (2014, 8th edition): Chapter 6, pp. 107-125

{7th edition, pp. 111-130}

Learning Checklist You should know what is meant by: Standardized scores z-scores You should be able to: Convert “raw scores” to standardized scores Transform scores from a specified normal distribution into z-scores Convert z-scores to scores on a scale with a specified mean and standard deviation Use tables to find the proportion of scores from the standard normal distribution that fall above, below, or between specified z-values.* [see e.g. 5 in the lecture] Find the proportion of scores from a normal distributed variable of known mean and standard deviation that fall above, below, or between specified values.* [see e.g. 6] {*You should also know that this is also the probability of randomly sampling a score from this range of values.} Use tables to find (approximate) values that specified proportions of scores within a normal distribution fall above or below. [see e.g. 7] Find symmetrical limits about the mean that specified proportions of scores in a normal distribution fall outside (‘extreme’ values) or between (‘central’ values). [see e.g. 8]

Weekly Exercises (1) After Lecture 3: Howell, pages 125-127, Questions 6.1, 6.2, 6.3, 6.12, 6.13(a) {see pages 130-132 in the 7th edition for these questions}

(2) After lecture 4a: Howell, pages 126-127, Questions 6.5*, 6.7, 6.9, 6.11, 6.18 {7th edition, pages 131-132, Questions 6.5*, 6.7. 6.9, 6.11, 6.14} Answers for odd-numbered questions are on pages 619-620 for the 8th edition {page 625 for the 7th edition} * The answer given in the textbook for Exercise 6.5c is incorrect (see below for the correct answer) Answers for even-numbered questions are below

Also try: http://www.uvm.edu/~dhowell/fundamentals7/SeeingStatisticsApplets/NormalDist.html

Answers for even-numbered questions and for Exercise 6.5c 6.2

6.5c

Using the population mean and standard deviation: For X = 2.5, z = –0.95 (just less than 1 standard deviation (SD) below the mean) For X = 6.2, z = +1.39 (about 11/3 SDs above the mean) For X = 9, z = +3.16 (about 3 SDs above the mean – ‘unusually high’) 62.7 or 62.6 (either answer is fine)

6.12

Convert initial scores to z-scores. Then convert z-scores onto a new scale with a mean of 80, and a SD of 10.

6.14 {7th ed., 6.18} The key point here is the distinction between sample statistics and population statistics. The mean Tscore of 50 and SD T-score of 10 refers the population mean and population SD. In other words, when the behavior problem scale was devised, a very large number of participants would have been assessed to determine that: the population mean = 50, and the population SD = 10. Figure 6.3 shows a sample of scores (sample mean = 49.1, sample SD = 10.6). The difference between the sample statistics and the population statistics may be due to sampling variability (a random sample from a population will not necessarily reflect all properties of the population perfectly). Another additional factor could be that the sample may not be a random sample from the population of scores that were used to determine the population mean and SD (e.g., perhaps some characteristics of this sample of adolescents make their average score slightly different to the overall average score). -9-...