Stats Test #2
(Samples and Population): Will a sample mean exactly = the mean of the population from which it was drawn, even if a treatment has had n effect.
No. It will rarely occur.
Key Concepts: What is the formula for Standard Error of the Mean for Population and sample? What category is it in?
Population: Not Applicable Sample: σM = σ f√ n Considered Variability
What is the Z-score for sample mean for the population and sample? What category is it in?
Population: Not applicable Sample: Z = (M - μ) / σM Considered Standardized scores
Key concepts: What is the Z-score formula for population and Sample? What category is it in?
Population: Z = X - μ σ Sample: Z = X - M s Considered- Standardized Scores
What are the formulas of Z-scores for observations from a sample.
Population= z = X - μ/ σ Sample = z = X - M/s
What is the relationship between z, X, μ and σ?
z = (X - μ) /σ X = μ + zσ μ = X - zσ σ = (X - μ)/ z z = (X - μ) z = (M - μ) σ σM X = μ + zσ M = μ + zσM μ = X - zσ μ = M - zσM σ = (X - μ) σM = (M - μ) z z
The Impact of sample size on the standard error
(assume the population standard deviation = 10)
True or False: In most situations, researcher would like the hypothesis test to reject the null hypothesis
T
True or False: It is possible for a very small treatment effect to be a statistically significant treatment effect
T
True or False: Sample size has little or no influence on measures of effect size
T
True or False: The alpha level determines the risk of a Type I error
T
True or False: The critical region for a hypothesis test consists of sample outcomes that are very unlikely to occur if the null hypothesis is true
T
True or False: The mean from the distribution of sample means is always equal to the mean for the population from which the sample was obtained
T
True or False: The value obtained for Cohen's d does not depend on the sample size
T
True or False: Two samples probably will have different means even if they are both the same size and they are both selected from the same population
T
True or False: If the power of a hypothesis test is calculated to be .80, then for the same test the probability of a Type II error is .20
T
The part that is shaded in on the normal distribution would be considered what?
The Mean to Z area of the distribution
True or False: As the population standard deviation increases, the standard error of the mean will also increase
T
True or False: Increasing the sample size increases the size of the Standard Error of the Mean
F
True or False: The power of a hypothesis test is the probability that the sample mean will be in the critical region even if the treatment has no effect
F
True or False: If other factors are held constant, lowering the alpha level will increase the power of a hypothesis test
F
Distribution of sample means: What is the CENTRAL LIMIT THEOREM?
For any population with μ and σ, the distribution of sample means for sample size n will have a of μ and σ / √n and will approach a normal distribution as n approaches infinity In this case infinity equals 30 or more - the distribution of sample means approaches a normal distribution once this n is reached.
What is the Inferential stats, it's use with z-scores, and its main concepts?
-Infer from sample data what the pop might think, Or to make judgements of the probability that an observed difference between groups is a dependable one or one that might have happened by chance in this study. - Z-score close to 0 have a high probability of occurring - it wouldn't be unusual to find a sample mean with that z-score if the treatment had no effect. If the z-score of the sample mean is close to that of the population mean it's unlikely the treatment had an effect. However, if the z-score is far from 0 it's much more likely that the treatment did have an effect. The probability of getting a sample mean far from the population mean just by chance is quite low. If the z-score of the sample mean is far from the population mean it's likely the treatment had an effect that leads to a change in the sample. B/c we know the prob of finding each z-score value in a normal distribution, we can use this info to better define 'highly unlikely'.
Some stat truths
1) Error is bad It's always desirable in statistics to decrease the amount of error that we have in our data - error adds confusion to the data and makes it more difficult to see the underlying patterns 2) Good things happen when you increase sample size Higher sample sizes lead to... - more stable data (ie., the addition of an extreme score doesn't have a big impact) - data that more closely resembles the underlying population from which it was drawn - data that contains less error than data that comes from smaller sample sizes
Example: Find the area under the standard normal curve above -1.98. Draw a picture and shade the appropriate proportion. List the steps to Solve.
1. Find the Proportion in Body column in the Unit Normal Table 2. Since we know we are looking at the body or larger portion. 3. The area is equal to 0.9761. Draw the picture with -1.98 and shade to the right of the score.
Finding z-scores from Areas: What z-score marks the top 50% of the distribution? What are the steps.
1. Look in either the Larger Portion of Smaller Portion column for .500 2. Look for the z-score that corresponds to that value 3. Z = 0
Finding Z-scores from Areas: What z-score marks the top 35% of the distribution? What are the steps.
1. Look in the Smaller Portion column for .35 2. Look for the z-score that corresponds to that value 3. Z = .39
Find Areas under the standard Normal Curve: Sketch the standard normal curve and shade the appropriate area under the curve for Z= 1.23. What are the steps to solve.
1. To find the area below z if z is above the mean, find the area that corresponds to z in Proportion in Body (Column B, or the larger portion) in the Unit Normal Table. 2. To shade, would be the area to the left of z of 1.23 = 0.8907.
In the Standard Normal distribution, the proportion of scores that fall between various z-scores / standard deviations are what % that fall with in the σ of the mean.
68% of cases fall within 1 σ of mean 95.5% of cases fall within 2 σ of the mean 99.7% of cases fall within 3 σ of the mean Standard deviation above and below the mean
1. A Type I error is when: a. We conclude that there is a meaningful effect in the population when in fact there is not. b. We conclude that there is not a meaningful effect in the population when in fact there is. c. We conclude that the test statistic is significant when in fact it is not. d. We conclude that there is no difference between the sample mean and the population mean
A
6. Which of the following best describes this sentence: 'Sleep deprivation will reduce the ability to perform a complex cognitive task'. a. A directional hypothesis b. An operational definition c. A null hypothesis d. A non-directional hypothesis
A
3. What is the conventional level of probability that is often accepted when conducting statistical tests? a. .1 b. .05 c. .5 d. .001
B
What is the reason of the standard deviation?
Bc its has been determined the proportion of scores that fall above and below any score in the standard normal distribution.
8. Which of the following statements would be considered a two-tailed hypothesis? a. A female doctor will be more empathetic than her male counterpart. b. As rates of immunization for measles increase, the incidence of measles will decrease. c. It is clear that there is a relationship between a healthy balanced diet and feelings of well-being d. The use of antibiotics will not impact the progression of a viral infection.
C
7. The significance of probability is usually expressed as a value occurring somewhere between 0 and 1. Which of the following would be considered the most highly significant in statistical terms? a. .1 b. .05 c. .025 d. .01
D
True or False: A Type II error occurs when a researcher concludes that a treatment has an effect when in fact the treatment has no effect
F
True or False: A researcher administers a treatment to a sample from a population with μ = 60. If the treatment is expected to increase scores and a one-tailed test is used to evaluate the treatment effect, then the null hypothesis would state that μ ≥ 60
F
True or False: A researcher is evaluating a treatment that is expected to increase scores. If a one-tailed test with alpha = .05 is used, then the critical region consists of z-scores less than -1.65.
F
True or False: A sample of n = 25 scores is selected from a population with a mean of 80 and a standard deviation of 20. The standard error for the sample mean is 20.
F
True or False: A treatment effect that is statistically significant will, by definition, be a meaningful difference.
F
True or False: If a hypothesis test fails to reject the null hypothesis, it means that the sample data failed to provide sufficient evidence to conclude that the treatment has an effect
F
True or False: If the sample data are in the critical region with alpha = .05, then the same sample data will be in the critical region if alpha were changed to .01
F
True or False: You can reduce the risk of a Type I error by using a larger sample
F
In the Standard Normal distribution, the proportion of score 68% of cases fall within What σ of the mean?
It falls within 1 σ of mean.
In the Standard Normal distribution, the proportion of score 95.5% of cases fall within What σ of the mean?
It falls within 2 σ of mean.
In the Standard Normal distribution, the proportion of score 99.7% of cases fall within What σ of the mean?
It falls within 3 σ of the mean.
Inferential Statistics and z-scores: What is the ultimate goal in inferential stats and how does Z-scores contribute to the ultimate goal?
It is to make generalizations about the population based on a sample or samples drawn from that population. It can help contribute by detecting a noticeable difference between population and sample mean. This can be done by comparing both means, by turning the sample mean into a z-score. The Z-score will compare sample mean to the population mean. Which can then determine the probability of getting that z-score, and use the probability value to draw conclusions.
The Z-score for Sample Means
It's possible to use a z-score to describe the exact location of any specific sample mean within the distribution of sample means. The z-score tells us where the sample mean is located in relation to all of the other possible sample means that could have been obtained The z-score formula for sample means is Z= M - μ σM The formula for the z-score for a sample mean taken from a population is very similar to the z-score for an observation taken from a population
Q4. What z-score marks the bottom 61% of the normal distribution?
Look in the Larger Portion column for the value closest to .61 We find the value .6103 2) Look across to the first column for the z-score that corresponds to that value The value .6103 corresponds to a z-score of .28 body shaded with the right tail un-shaded
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the probability that a randomly selected student's score on the test is less than 28
Step 1: Find the z-score for the value of in this distribution Z = (28 - 31.2)/5; Z = -3.1/5; Z = -.62 Step 2: Determine what % of scores fall below the z-score found in Step 1 (z = -.62) Look in Smaller Portion column in Appendix z for the z-score of .62 p = .2676 = .27 (The probability is .27 that a score is below 28, ie., 28% of scores in this distribution have a value below 28)
In the __________________ ________________ , the scores in the pop or sample follow a normal distribution, it can be determined the % of scores that fall above or below any value in the pop or sample, once it has been converted that value to a Z- score.
The Standard Deviation
On the Appendix Z: What is the smaller proportion of the distribution is called?
The Tail.
On the Appendix Z: What is the larger portion of the distribution is called?
The body.
On the Appendix Z: The Mean to Z area of the distribution would be recognized on the bell curve as what?
The part that would be shaded in.
How do you know when to find the area between 2 scores, 1 positive and 1 negative Z in the normal distribution? Where would you look to find your answer?
When finding areas between 2 scores, 1 positive and 1 negative, you know to find the portion between mean and Z (as well as adding them up) because your finding Z-scores from both sides. You know where to look because your finding the between section.
Are the concepts the same for population and sample Z-scores?
Yes it is the same for both.
If the area is between 2 scores, 1 positive and 1 negative Z between the mean. Which direction would you shade in the normal distribution and why?
You would shade in between the 1st Z-score and the 2nd Z-score because your looking at the between scores so you would shade in the middle (or between).
If the area is below Z, then the Z is above the mean. Which direction would you shade in the normal distribution and why?
You would shade left of the BODY because the Z is above the mean .
If the area is to the right of Z, then the Z is above the mean. Which direction would you shade in the normal distribution and why?
You would shade to the right of the TAIL because the Z is above the mean and it is considered the smaller portion.
What is a Sampling Distribution?
is a distribution of statistics obtained by selecting all of the possible samples of a specific size from a population The Distribution of Sample Means is one example of a sampling distribution
What is the Distribution of Sample means?
is the collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population
The formula for the Standard Error of the Mean is:
σM = σ n of size n is equal to the population standard deviation divided by the square root of the size of the samples that make up that Distribution of Sample Means Essentially, the sample means for all samples, regardless of size, starts off the same - it's equal to the standard deviation of the population from which the samples were drawn Then the size of the sample from which the means were drawn comes into play Dividing the population standard deviation by the square root of n directly impacts it. - as the n increases, the square root of n increases, which means it (M) decreases - e.g., dividing the numerator (the standard deviation) by 4 decreases it more than does dividing it by 2
Illustrating the Distributions of Sample means
We take a random sample of 8 observations from the population and compute the mean of tht sample each bar represents an observation The mean of these 8 observatiosn is 21 We do this again and again, selecting different groups of 8 observations each time, until we've collected the means of all possible samples where n = 8
What does the formula for Z-Score for Sample Means tell You?
z = M - μ σM The sign (+/-) of the z-score tells you if the sample mean is above or below the population mean The value tells you how far above or below the population mean the sample mean is M - μ- means How far is the sample mean from the population mean... σM- means stated in number of Standard Error of the Mean units Example: μ = 72, M = 76, σM = 2 z = (76-72) = 2 2
Find the area to the right of Z, if Z is above the mean, Sketch the standard normal curve and shade the appropriate area under the curve for Z= 1.23. What are the steps to solve.
1. Use the Unit Normal Table to find the Proportion in Tail (Column C, Smaller portion) that corresponds to z. 2. To shade, would be the proportion in Tail for z of 1.23 = 0.1093. (You could also that it would be the right tail, since it is positive).
Find the Area Between Two Scores, one positive and one negative: Sketch the standard normal curve and shade the appropriate area under the curve for Z= -0.75 and 1.23. What are the steps to solve.
1. You would find the Proportion Between Mean and z corresponding to each z-score in the Unit Normal Table and add them together. 2. Find the proportion Between Mean and z for z of -.75 = 2734 3. Find the proportion Between Mean and z for z of 1.23 = .3907 4. Then take both portions between the mean and Z and add together, .2734 + .3907 = .6641 5. You would shade between -0.75 and 1.23
When finding Areas of the curve, What % of scores fall below Z= -1.00? What are the steps you would take to solve.
1. You would look in the Z column in the Appendix Z, under where it says Z=-1.00. 2. Then to find your answer you would look in the same row under the larger portion and find your answer of .8413 and round to Z=.84 % It would be shaded to the right of -1.00.
When finding Areas of the curve, What % of scores fall below Z= 1.28? If you had to shade it on the Normal Distribution, Where would it be shaded at? What are the steps you would take to solve.
1. You would look in the Z column in the Appendix Z, under where it says Z=1.28. 2. Then to find your answer you would look in the same row under the Mean to Z and find your answer of .3997 and round to Z= .40 % It would be shaded to the left of 1.28.
When finding Areas of the curve, What % of scores fall between the mean and Z= 1.5? What are the steps you would take to solve.
1. You would look in the Z column in the Appendix Z, under where it says Z=1.5. 2. Then to find your answer you would look in the same row under the smaller portion and find your answer of .1003 and round to Z=.10% It would be shaded in between 0 and 1.28.
SAMPLING ERROR is what? Why does this happen? Because of the sample error we expect that________________1__________________________________. We need to understand how _________2______ we can __________________________________ to be from the population mean just by __________ in order to determine how ________ or _______ any particular sample mean is. Fortunately the data set of all possible sample means taken from a population form a _________ & ____________ pattern. This patterns allows us to what? The ability to make predictions about any sample is based on our knowledge of the _____________ _______________ _________ __________.
1. because we can't gather data from everybody in our sample, and so differences will occur. 2. each sample mean will differ to at least some extent from the population mean. 3. different; expect our sample mean; chance; likely; unlikely. -simple and predictable. - make accurate predictions about the likelihood of getting any particular sample mean taken from the population of interest. -Distribution of Sample Means.
Example: Find the are under the standard normal curve to the left of Z= -2.33. Draw a picture and shade the appropriate proportion. List the steps to Solve.
1. find the Z score in the Appendix Z 2. Then since we already know its to the left (neg) we are going to look at the proportion in the TAIL column. 3. The answer is the area is = to .0099 Draw the picture with -2.33 shaded in to the left or (the Tail shaded in).
A null hypothesis: a. States that the experimental treatment will have an effect. b. Is also called "the researcher's hypothesis" c. Predicts that the experimental treatment will have no effect. d. None of the above.
C
2. Which of these statements about statistical power is not true? a. Power is the ability of a test to detect an effect. b. We can use power to determine how big a sample is required to detect an effect of a certain size. c. Power is dependent at least in part to the probability of making a Type I error. d. All of the above are true.
D
5. Which of the following terms best describes the sentence: 'In a blind tasting, people will not be able to tell the difference between margarine and butter'? a. A directional hypothesis b. An operational definition c. A null hypothesis d. A non-directional hypothesis
D
True or False: In a hypothesis test, the likelihood of rejecting the null hypothesis does not depend on the sample size
F
True or False: In order for the distribution of sample means to be normal, it must be based on samples of at least n = 30 scores
F
How do you know when to find the area below Z in the normal distribution? Where would you look to find your answer?
If Z is above the mean, then you would find your answer in the larger portion.
How do you know when to find the area to the right of Z in the normal distribution? Where would you look to find your answer?
If Z is above the mean, then you would find your answer in the smaller portion because it is considered the TAIL.
How do we use the standard error of the mean?
Just as we use the mean and standard deviation to describe the distribution of individual scores in a population or sample, we use the mean and standard error of the mean to describe the distribution of sample means Population Described by Population Mean and Standard Deviation Distribution of sample means Described by Population Mean and Standard Error of the Mean
Q3. What z-score marks the top 74% of the normal distribution?
Look in the Larger Portion column for the value closest to .74 We find the value .7389 2) Look across to the first column for the z-score that corresponds to that value The value .7389 corresponds to a z-score of .64 Since the value is on the left side (below the mean) of the distribution, the z-score is negative. Z = - .64 The body shaded in with the left tail un-shaded
Q2. What z-score marks the bottom 23% of the normal distribution?
Look in the Smaller Portion column for the value closest to .23 We find the value .2296 2) Look across to the first column for the z-score that corresponds to that value The value .2296 corresponds to a z-score of .74 Since the value is on the left side (below the mean) of the distribution, the z-score is negative. Z = - .74 shaded on the left tail
Finding the Z-scores from Areas: Q1. What z-score marks the top 35% of the normal distribution?
Look in the Smaller Portion column for the value closest to .35 We find the value .3483 and .3520 2) Look across to the first column for the z-score that corresponds to that value The value .3483 corresponds to a z-score of .39 The value .3520 corresponds to a z-score of .38 Either is acceptable Shaded on the right tail
True or False: If all other factors are held constant, then increasing the sample size will increase the likelihood of rejecting the null hypothesis
T
True or False: In a research report, the term significant result means that the null hypothesis was rejected
T
True or False: In general, the null hypothesis predicts that the treatment has no effect on the population mean
T
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the probability that a randomly selected student's score on the test is between 30 and 34
Step 1: Convert the two X values into z-scores Step 2: Find the value of Mean to Z for the z-score for X = 30 Step 3: Find the value of Mean to Z for the z-score for X = 34 Step 4: Add the values of Step 2 and Step 3 Step 1: Convert the two X values into z-scores Z = (30 - 32.2) / 5; z = -2.2/5; z = -.44 Z = (34 - 32.2) / 5; z = 1.8/5; z = .36 Step 2: Find the value of Mean to Z for the z-score of -.44 Mean to Z for z = .44 = .17 Step 3: Find the value of Mean to Z for the z-score of +.36 Mean to z for z = .36 = .1406 Step 4: Add the values of Step 2 and Step 3 .17 + .1406 = .3106 p = .3106 = .31 (The probability is .31 that a score is between 30 and 34; ie., 31% of scores have a value between 30 and 34)
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the probability that a randomly selected student's score on the test is greater than 37
Step 1: Find the z-score for the value of 37 in this distribution Step 2: Determine what % of scores fall above the z-score found in Step 1 Step 1: Find the z-score for the value of 37 in this distribution Z = (X - μ) / σ Z = (37 - 31.2) / 5; Z = 5.8 / 5; Z = 1.17 Step 2: Determine what % of scores fall above the z-score found in Step 1 (z = 1.17) Look in "Smaller Portion" column of Appendix z for the z-score of 1.17 P = .121 = .12; The probability of getting a score higher than 37 is .12 (12%)
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the z-score that corresponds to the top 5% of the scores in the distribution
Step 1: Look for the value closest to .05 in the Smaller Portion column of Appendix z The values .0505 and .0495 come closest Step 2: Find the z-score that corresponds to the value in Step 1 The z-score for .0505 is 1.64 and the z-score for .0495 is 1.65. Either is acceptable
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the raw score that corresponds to the bottom 9% of the scores in the distribution
Step 1: Look for the value closest to .09 in the Smaller Portion column of Appendix z Step 2: Find the z-score that corresponds to the value in Step 1 Step 3: Find the raw score (M value) that corresponds to the z-score in Step 2 Step 1: The value closest to .09 in the Smaller Portion column is .0901 Step 2: The z-score that corresponds to Smaller Portion = .0901 is 1.34 Because the z-score we are looking for is below the mean our desired z-score becomes -1.34 Step 3: Find the raw score that corresponds to z = -1.34 X = μ + zσ (see last page in this deck for this formula) X = 31.2 + (-1.34)(5) X = 31.2 - 6.7 X = 24.5
Assume the mean of a test is 31.2 and the standard deviation is 5 Find the raw score that corresponds to the top 10% of the scores in the distribution
Step 1: Look for the value closest to .10 in the Smaller Portion column of Appendix z Step 2: Find the z-score that corresponds to the value in Step 1 Step 3: Find the raw score (M value) that corresponds to the z-score in Step 2 Step 1: The value of .1003 is the closest to .10 in the Smaller Portion column Step 2: The z-score of 1.28 corresponds to the value from Step 1 Step 3: Find the raw score (M value) that corresponds to z = +1.28 X = μ + zσ (see last page in this deck for this formula) X = 31.2 + (1.28)(5) X = 31.2 + (6.4) X = 37.6
Find the z-score that corresponds to the bottom 22% of the scores in the distribution
Step 1: Look for the value closest to .22 in the Smaller Portion column of Appendix z Step 2: Find the z-score that corresponds to the value in Step 1 Step 1: Look for the value closest to .22 in the Smaller Portion column .2206 is the closest value to .22 in the Smaller Portions column Step 2: Find the z-score that corresponds to the value in Step 1 The z-score for .2206 in the Smaller Portions column is .77 Since the z-score we want is below the mean this becomes -.77
True or False: A Type I error occurs when a treatment has no effect but the decision is to reject the null hypothesis
T
True or False: If all other factors are held constant, increasing the sample size from n = 25 to n = 100 will increase the power of a statistical test
T
What is the Standard Error of the Mean (Standard Error of M)? What's its symbol?
The measure of standard deviation for the Distribution of Sample Means. - provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ) -The symbol is σM - has two components: 1. The σ of the population from which the M were drawn 2. It makes sense to take this into consideration since samples drawn from a population whose observations are spread far apart will have means that are also spread farther apart compared to those that are drawn from a population whose observations are relatively close together. -which will lead to a greater spread in the distribution of M for the 1st population vs the 2nd population. -Similarly, the means of samples made up of randomly chosen values from the top population will be more spread out than the means of samples made up of randomly chosen values from the bottom population - A sample mean made up of extreme values from this population will be greater than a sample mean made up of extreme values from this population. -The size of the samples that make up the Dist. of M. In other words, as sample size (n) increases, the size of the sample error decreases (Larger samples are more accurate representations of the population)
Sample Means / Sampling Distribution of the Means • On average, what do we expect the sample mean to be equal to?
The population mean. We know that because of sample error this is rarely the case but, in the absence of any other information we assume the sample mean is equal to the population mean
How do we use the distribution of sample Means?
The primary use of the distribution of sample means is to find the probability associated with any specific sample. This allows us to answer questions such as... How likely is it to that we would find a sample mean of this value or higher? How likely is it to that we would find a sample mean of this value or lower? What statistic have we already learned that will allow us to answer questions like this? The Z-score
What is Sampling error?
The random difference between a sample mean and the mean of the population from which it was drawn.
The distribution of sample means
The sample means form a Distribution of Sample Means In order to draw conclusions about a sample based on the distribution of sample means, it must contain all possible samples of that size taken from the population - if it doesn't we can't know the probability of obtaining any specific sample value Also notice that we now have a distribution of means In other words we have a distribution of statistics rather than a distribution of observations.
Distribution of Sample Means: Some characteristics of the distribution of sample means
The sample means should cluster around the population mean. Because of sampling error we don't expect that each will be exactly equal to the population mean but they should be relatively close to it, since most of the observations in the population are relatively close to the population mean (remember, 68% are within one SD of the mean) 2. The sample means should form a normal-shaped distribution If the underlying population mean forms a normal distribution the distribution of sample means will be normal Even if the underlying population does not form a normal distribution, we expect our distribution of sample means will be normally distributed if our sample size is 30 or higher We know this from the CENTRAL LIMIT THEOREM 3. The greater the number of observations in the sample, the closer the sample mean should be to the population mean A larger sample should be more representative of the population than a smaller sample More population subgroups will be represented in the sample The sample means obtained from large sample sizes should cluster closely around the population mean, while the means from samples with small n's should be more scattered One or two extreme values taken from the population have a bigger impact on a smaller sample than on a larger sample
Distribution of Sample means
We know that in an ideal (statistical) world there would be no distribution of sample means, there would only be one value Every sample we draw would have the same mean --the mean of the population We also know that this isn't the case because of sampling error [which you'll remember is defined as the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter] Look at these two distributions of sample means Assuming each has the same total # of sample means... 1) which one has more total error? 2) which one has a higher n in the samples that make up that distribution? The total amount of error for any Distribution of Sample Means reflects how far, on average, each sample mean in the distribution differs from the true -ie., population -mean What tells us how far, on average, the scores in a distribution differ from the mean? The Standard Deviation
