*Homework discussion:* Z-scores

Hello, everyone. Below are some submitted answers to the homework covering material from the z-scores lecture, as well as my commentary. Feel free to ask questions in the comments section below.

As always, there are many ways to say the same thing and many ways to say correct things!

*With all lecture audio, please be careful about your computer volume and your use of headphones or earbuds.*

**Join the discussion in the Comments section. What answers are better than others and why? Feel free to specify Question # and Response # when you post (Q#R#)!**

**(1) Using SPSS, report in the form below the mean, mode, median, variance, and standard deviation. Make sure to name and label your variable in SPSS. Save your file, as well, in case you need to access it later. (You’d hate to have to reenter everything!)**

Please watch the following video. Did you make sure to name and label your variable? It’s also a good idea to indicate that your variable was “scale”, meaning essentially interval-ratio. The other options are more straightforward. If you have a file with dozens of columns of data, it’s a good idea to keep track of the types of variables you have.

Please make sure to report your descriptive statistics to two decimal places. Thus, the standard deviation is 2.49 and the variance is 6.19. (Note the superscript “a”.) There are multiple modes! Do I have to find the other mode or modes by hand? What’s the point of SPSS? Here’s how to do it.

1. Go to Analyze → Descriptive Statistics → Frequencies…

2. Choose your variable. (Note below that I also have the option of my newly created *z*-score ‘variable’.

3. Select “Display frequency tables”. (If you go to Analyze → Descriptive Statistics → Descriptives…, that box would be how you get your *z*-scores, remember.)

4. Hit ok.

Here is your output table. Note the mode of “12” and the other mode of “13”. Also note the bunching around 11, 12, 13. Consider the percentages associated with these values. Note also that by the time you get to “8” and “17” and “18”, the frequencies and percentages are quite a bit lower.

**(2) How many standard deviations was your observed number of punches from the mean number of punches for our class? (Always report z-scores to two decimal places.)**

This will depend on your observed punches, but make sure to indicate in the future both your score and your *z*-score, the latter of which requires two decimal places and a zero preceding the decimal place if it’s less than zero.

**(3) Above we have a sample. Given this sample, what is the z-score for the mean of our population? Tell why with as much detail as is necessary to demonstrate your mastery of these concepts. Make sure to explain to me what a z-score is. (Do so in a way that does not involve writing out a formula. A formula is not an explanation.) Conceptually, what is a z-score?**

- 3. Z- Score for population mean= 0 , because z-score is a measure of how many standard deviations away (above or below) from the mean a score is . The mean is O standard deviations from itself.
- 3. A z-score is a number that tells you how far away a number in a group of numbers is from the mean of that group. A z-score can be positive or negative, meaning if it is positive it is above the mean and negative would mean it is below the mean. The z-score for the mean of our population was .71. How was this calculated. It came out of nowhere!
*Z*-scores below zero require a “0”, so 0.71. - 3. A z-score allows a comparison across distributions. Meaning, that if we have this class setting the standard deviation at 2.488 [use two decimal places], and I had a z-score of .50. How was that calculated? Then, we could compare my z-score with someone from a different class because z-scores are standard units of deviation. We’re interested in the mean of the population given the mean of the sample. The z-score for every mean in any sample would be 0, as well.
- 3. For a population, we want to generalize the findings of the sample. That’s a confusing first sentence. We want to generalize the findings of a sample
*to the*population, yes. The mean of the sample was 12.76 so we could generalize that to the population as well. Yes. In fact we report the mean of the sample as an estimate of the mean of the population from which it came. We don’t need to change the formula. The variance, as you know, is a different story. The z-score for the mean of our population then would be 0 as a z-score represents how far a certain score is away from the mean. If the score in question is also the mean, it would be 0 away from the mean and thus the Z-score would be 0. A Z-score is a standardized score that helps the researcher understand how that score compares with the rest of the scores and can also be compared with other variables. Otherwise great. - 3. A z-score is a ratio that tells us how far away a particular score is from the mean. Well, I’m glad you mentioned it was a ratio, but the rest of that first sentence just describes the numerator of that ratio. You described a deviation score, not a
*z*-score. It does this in units of standard deviations. Yes! Don’t break it up into two sentences. It makes the first sentence misleading or wrong and the second sentence use words like “it” and “this” which can be vague as to that to which they refer. This question is asking for the z-score of the mean, and since the mean is (obviously) numerically equivalent to the mean, it’s z-score would be 0. It is directly on the mean, so therefore its distance from the mean is 0. Great. - 3. A z-score is simply the measurement of a value’s relationship [what does relationship mean?] to the mean; z-scores can be positive or negative. This indicates whether the value is above or below the mean and by how many standard deviations [how many standard deviations what?].For example, a z-score of 0 indicates that the z-score is equal to the mean, a z-score of -1 indicates that the z-score is 1 standard deviation below the mean, and a z-score of +1 indicates that the z-score is 1 standard deviation above the mean. Therefore, the z-score for the mean of our population would be 0 [because it’s not numerically different from itself] Just say more here and there and you’ve got it.
- 3. Given this sample, the z-score for the mean of our population is zero. This is because the z-score for any distribution is zero [that’s not what you mean] because the z-score is the measure of the distance of an observation from the mean. The mean is no distance from itself, so the z-score is zero. Conceptually, a z-score is a type of standard score that tells us how far away a score is from the mean in standard deviation units. It tells us by how many standard deviations the corresponding raw score lies above or below the mean of distribution. Great.
- 3. A Z-score is telling us how far away the score is from the mean in standard deviation units. For example it lets you standardize scores so you could compare the ACT to the SAT off of the bell curse. If we are using the first example, the mean being 100 and the Std. deviation being 120 [20?], what it does not tell us is the number of people who are observing or taking part in this (which would be the X) [the number taking part is the
*n*? not the*x*, which is just an observation] A z-score of + 1 means that a score of 120 is one standard deviation about the mean of that distribution. (You look at what is between 100-120 on the bell curse, the raw score would be equal to a z-score of 120 + 1) Which means what? I can’t tell if you are answering the question or not. We are interested in the*z*-score of the mean of the population. - 3. To find the z-score using the sample above, we would take the sample [sample what? you can’t take a “sample” as a distribution of observations and do something mathematical to it] minus the [sample?] mean then divide that by the standard deviation. 13-12.67= 0.33/ 2.582= .1278079009 and round it. z=0.13 The question asks to define conceptually what a
*z*-score is. Had you done this, a mean being 0.13 standard deviations from itself would strike you as odd. Use each component of a HW question to assist you on the others as clues. - 3. [no response] Don’t leave anything blank! I can’t offer any comments and you get no practice if you leave anything blank!
- 3. Zscores help us understand raw scores and where they are in relation to the mean. [Howso? How is that different from deviation scores?] This in turn helps us began to make inferences about our data and explain it in a way that makes sense using percentages and probability in terms of the relationships between the variables But you didn’t answer the actual homework question.
- 3. The z-score for the mean of the population is 0. This is because a z-score measures distance from the mean (in units of standard deviation), thus if you are looking right at the mean the z-value would be 0. A negative score means the value lies below the mean and a positive score means the value lies above the mean. The Main difference between standard deviation and a z-score is that standard deviation reflects the variability of a sample or population. It creates a point of reference to which a z-score can use like a unit to convey where a certain value would lie on a normal distribution curve. This second part is nice but I’m not sure you’re asked to distinguish between a standard deviation and a z-score.
- 3. The z-score tells us how far a score is from the mean in standard deviation units. This would make the z-score of a mean: zero. A mean cannot be any unit away from itself.
- 3. A z-score is a type of standard score which allows us to make raw scores meaningful, since scores by themselves don’t have any meaning without context. [I know what you mean here. Perhaps try to explain then what a z-score provides in the way of meaning]
- 3. A z-score describes a score in terms of how much it its above or below the mean.[This just sounds like a deviation score] It is the number of standard deviations that score from a given sample is above or below the mean for that sample so it is a score that has been transformed so that is can better show the score’s location in the distribution. Better with respect to what? Both a deviation score and a
*z*-score tell you about distance from the mean, but they just do so with different kinds of units. - 3. The z-score for the mean of our population is zero. A z-score measures the relationship between a score and the mean. [Consider that second sentence is isolation. What does it mean? Try to avoid sentences like that because you know what you’re saying.] If a z-score for the mean is calculated, then the z-score is always zero as it is zero standard deviations away from the mean because the score being analyzed is the mean itself.
- 3. The z score for the mean of our population would be zero because a z score desccribes distance from the mean (more specifically it shows how many standard deviations a score is from the mean) [the part in parentheses is better so why not just say that instead of ‘describes distance from the mean’]. So the mean itself would not have any distance from the mean.
- 3. The z-score for the mean of the population would be 0.00. A z-score represents how far away a score is from the mean in standard deviation units. Thus, the z-score of any mean would be 0.00 because the z-score of the mean would not vary from the mean itself.
- 3. To find the population’s z-score, we would take a mean for another sample group’s data set and subtract our mean from that population’s mean and then divide that value by our standard deviation multiplied by the square root of the sample size. [Why are we multiplying by the square root of the sample size?] The z-score gives a standard score for raw data and gives a value for how far a piece of data [what is a piece of data? An observation? Just say that.] is from the mean in terms of standard deviations. When a population’s z-score is found [no such thing. a
*z*-score is something that can be determined for any value, but for a ‘population’] , it [what is ‘it’?] takes the data from a sample that is present and determines how far it is from the sample mean that it’s being compared to. [In that sentence it’s not clear what’s being compared to what. Break it down for simplicity. Stay away from words like “it”. Just say what it is directly.] Z-scores explain how many standard deviations the raw data points are away from the mean to give the raw data actual meaning and a reference point. Say more because you’re repeating things that elements of truth to them, but it’s not coming together.

*Please feel free to ask a question in the comments section below. They can be anonymous, but signing with your initials would be helpful for me!
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If our answers aren’t reflected below, should we be concerned our homework was not received or are only some of the responses listed?

I received it.

great. thank you.

Hi Dr. Ramey, I know that I submitted my answers very late, but I did get the data entered correctly and ran the descriptive statistics report correctly. I think I understand now how the z score differs from the raw score and the standard deviation. The z score is any scores relative distance from the mean as measured by the standard deviation – not it’s specific distance from the mean in an interval but as ration measured by that numeric value which represents the only valid way of measuring any distance between scores in a sample or population. If you have a frequency distribution ( 11 of the scores representing a sample of 29 when duplicates are compensated for). I think. Can’t believe I’m posting this potentially very dumb response info on the blog.

Thank you for the review! I had a quick question about the mode. So after I made the table in my SPSS file, I noticed the little footnote that stated there were multiple modes so I used the frequency table to find the other one. In the future would you want us to only report the smallest mode?

You would want to report all modes.

(SPSS is just funky here and there.)