Z SCORES FOR SAMPLE MEANS

CAVEAT

Hopefully everything from the Descriptive vs. Influential statistics module makes sense. If not, consider reviewing it again! We are going to take that information and combine it with what you know about standard deviation to move onward!

Let's start today with a video overview of what we will be discussing:

{VIDEO}

Introduction

In the lesson on Z Scores, we learned that Z Score is nothing scary--it just means "number of standard deviations from the mean". So, what if we have a dataset with a mean of 50 and standard deviation of 15. What is the Z Score of an observation in that dataset with the value 80? 

Take a quiz to see if you remember!

{Quiz. A: Z Score=2}

That quiz is a reminder that Z Score is nothing scary. It just means "number of standard deviations from the mean!" Sometimes we have to first figure out how far away something is from the mean, but it is otherwise quite straightforward!

"So, how can we combine this with inferential statistics?"

I'm glad you asked! We can do the same thing with sample means that we can with observations in a sample.

Observations in a Sample vs. Sample Means in a Distribution

^^^^ That subtitle is a little intense! Let's break it down into English. (As always, this is about translating from Statistics language to English--or really whatever language you are learning stats in!).

Examples usually help make the point:

EXAMPLE OF OBSERVATIONS IN A SAMPLE

This one should seem familiar. Suppose we have a classroom of 10 students (it is a small classroom). They took a stats test and got the following scores:

98, 92, 90, 95, 88, 94, 95, 100, 99, 98

(They are very good students)

If we want to know the Z Score of the first observation (98), what do we do? What we have been doing all along! 

Let's try again (just for fun!).

Start by adding up the observations and dividing by the number of observations:

Xi
98
92
90
95
88
94
95
100
99
98
94.9<--mean!

Now, subtract 94.9 (the mean) from each observation to get the difference of each observation from the mean (that is the heart and soul of variance!).

Xi	Xi-Xbar
98	3.1
92	-2.9
90	-4.9
95	0.1
88	-6.9
94	-0.9
95	0.1
100	5.1
99	4.1
98	3.1

Now, add up those differences. REMEMBER, THIS IS A CHECKPOINT AND SHOULD COME OUT TO ZERO, MEANING WE HAVE SUCCESSFULLY BALANCED THE SEESAW!

Xi-Xbar
3.1
-2.9
-4.9
0.1
-6.9
-0.9
0.1
5.1
4.1
3.1
-5.6843E-14

Wait! The mean is not zero, but -0.000000000000056843! 

All is well! In Stats language, we call this "rounding error". It means that we rounded to tenths (0.1) or hundredths (0.01) so our numbers are not exact. That is fine! As long as that number is tiny, we are still OK.

*The very aspiring and astute student will usually ask at this point, "How tiny is tiny?" As a rule of thumb, if this number is less than 0.01% of the mean, you are (probably) OK! So take your mean, multiply by .0001 and if the difference is less than that, chances are it is just rounding error!*

Next, square the differences so that the summed differences don't add up to only zero!

Xi	Xi-Xbar	diff^2
98	3.1	9.61
92	-2.9	8.41
90	-4.9	24.01
95	0.1	0.01
88	-6.9	47.61
94	-0.9	0.81
95	0.1	0.01
100	5.1	26.01
99	4.1	16.81
98	3.1	9.61
94.9	-5.6843E-14	142.9

So, we have summed differences equal to 142.9!

Now, how do we compute variance? Divide by the number of observations (n)!

142.9/10=14.29

14.29 = variance

How do we get standard deviation? S squared = S squared! Variance is S (standard deviation) squared! So, just take the square root of variance to get S (standard deviation). 

sq rt of 14.29=3.78! This is standard deviation! 

Now, to get the Z Score--the number of standard deviations from the mean--we need to know how far the observation of interest (98) is from the mean:

98-94.9=3.1

So, how do we find out how many standard deviations 3.1 is? The same way you find out how many feet 24 inches is--DIVIDE! 

3.1/3.78=0.82

So, 0.82 is the number of standard deviations the 98 is from the mean! We can translate it like this Z Score of 98=0.82. 

So, the Z Score is 0.82! 

EASY! (?)

EXAMPLE OF SAMPLE IN A DISTRIBUTION

Now, suppose we want to know the last stats exam score for every college student in America! It just became unfeasible to collect all of those scores! It will cost too much (time, money, etc). This means you will have to take the largest sample you can and hope it comes close to the "true" population average! 

OF UTMOST IMPORTANCE--> As the video shows, choosing the largest sample will give you the best chance of getting a sample average that matches the "true" population average. This is due to the "Law of Large Numbers"and the Central Limit Theorem (CLT)! More on these later...

Think back to the video now. Imagine 100,000 test scores (I won't put a column that big in this module, so your imagination will have to do!). Let us suppose that you can afford to do a sample of 100 students. You also have taken a stats class and know that you should take a random sample to give yourself the best chance at that sample average matching the "true" population average. 

Random=All element in the population have a non-zero chance of being chosen and all elements have an equal chance of being chosen. 

Because we are taking a random sample, you never know which 100 observations will be chosen. This means that there will be some variation in the sample's average compared to other possible samples. In the video, we used a simulator to show that a sample of a given size will be a little different each time if you take repeated samples. 

This means that (drum roll...) SAMPLE MEANS HAVE A DISTRIBUTION THEMSELVES!

So, now picture that we take 10 samples of 100 (10 samples of 100 observations each) and we record the average for each sample of 100. The first has an average score of 78, the second has an average of 72, the third of 70 and so on.

Here are the averages for all 10 samples of 100:

78, 72, 70, 75, 68, 74, 75, 80, 79, 78

THERE IS GOOD NEWS AND BAD NEWS!

The good news is that we can still compute Z Scores in the same GENERAL way we have been doing! The bad news is that we now have to make some adjustments because we are not collecting all the information in the population--we are trying to estimate it.

When we estimate, we must adjust.
When we estimate, we must adjust.
When we estimate, we must adjust. 

Let's first look at what we do exactly the same way as what we have been doing:

Means	Mean-Avg. of the means	diff^2
78	3.1	9.61
72	-2.9	8.41
70	-4.9	24.01
75	0.1	0.01
68	-6.9	47.61
74	-0.9	0.81
75	0.1	0.01
80	5.1	26.01
79	4.1	16.81
78	3.1	9.61
74.9	-5.68434E-14	142.9

Notice that there are slight modifications to the headings because we now have Means in column 1. We still take the average of that column just like always and it gets a little awkward in the wording because we now have a mean of the means...

To slip out of that awkward wording, we call it THE GRAND MEAN!

The Grand Mean!

Here it is again, with the new title:

Means	Grand Mean	diff^2
78	3.1	9.61
72	-2.9	8.41
70	-4.9	24.01
75	0.1	0.01
68	-6.9	47.61
74	-0.9	0.81
75	0.1	0.01
80	5.1	26.01
79	4.1	16.81
78	3.1	9.61
74.9	-5.68434E-14	142.9

Notice that everything up to this point is just as you are used to.

However, when we divide the sum of squared differences (142.9), we must make some of those adjustments!

When we estimate, we must adjust!

Because we took samples, and did not get information from the whole population, there is going to be some error in our variance and standard deviation if we just divide 142.9 by the number of means (10). This is where "n-1" comes into play! We have 10 means, but we can subtract 1 as an adjustment that will make the variance and standard deviation closer to the "true" population variance and standard deviation.

There will be a quiz in a minute, so be sure to take note:

FOR SD OF OBSERVATIONS IN A SAMPLE, DIVIDE BY N
FOR SD OF MEANS IN A DISTRIBUTION, DIVIDE BY N-1!

DESCRIPTIVE SD = DIVIDE BY N
INFERENTIAL SD =DIVIDE BY N-1!

So 142.9/10 gives variance =14.29
BUT 142.9/9=15.88

***Some of you will be asking, "How do we know that dividing by n-1 gives us a better estimate of the "true" population standard than dividing by just n." Here is proof:

Option 1

Option 2, think of this as more STATS MAGIC! 

If you have all elements of the population, divide by n, if not, divide by n-1!

Most students prefer Option 2...***

Now that we have 142.9/9=15.88, we know the variance.

We can take the square root and have standard deviation! sq rt of 15.88=3.98!

Now you try...

{quiz}

Hopefully that quiz went well. If not, don't worry, you will get more practice!

Means have a distribution

Some of you will have caught on by now, but sample means have a distribution just like observations in a sample! Remember the bell curve?

It still works! It is mostly a theoretical distribution, but if you took all possible samples of a given size, the sample averages would make this same shape! Most would be close to the "true" population mean, right in the center. Few would be extremely larger or smaller than the "true" population mean. But everything holds true for the average of samples in a theoretical distribution of all possible samples compared to observations within a sample. The standard deviation is computed (almost) the same way (just remember n-1 instead of n!), the standard deviations still correspond to their same respective area under the curve at each point. 

Now is a good time to memorize some of those main areas:

Area between -1 and 1=68.2%

Area between -2 and 2=95.4%

Area between -3 and 3=99.6%

Take a second to commit them to memory, then try a quiz!

{quiz}

How did that go? You will definitely be seeing more of those in the future so be sure to memorize them if you didn't get them correct on the quiz!

Population parameters:

The theoretical distribution of sample means has a standard deviation, but because it is a theoretical distribution, we almost never know what it "truly" is. We estimate it like this: standard deviation of your sample divided by the square root of the number in your sample. 

In stats language: s/(√n)

{quiz. How do you write: standard deviation of your sample divided by the square root of the number in your sample in Stats language?}. 

Great! This standard deviation also has a special name "standard error"

Standard error=standard deviation of the theoretical distribution of all possible samples≈standard deviation of your sample divided by the number in the sample=s/(√n)

≈ (mean that it is roughly equivalent. In this case, it is an estimate!)

Conclusions

• When we have a really big population, and it is not feasible to get information from every single person, we can take a sample and use it to estimate the true population mean!
• We can take a sample of a given size and compute the mean of that sample in the same way we have been computing the mean all along!
• The larger the sample, the more likely it will be to be a close estimate of the "true" population mean.
• We can compute Z Scores for sample means the same way we computed Z Scores for observations in a sample--the only difference is that we divide the sum of squared variance by n-1 instead of just n. That helps make it a more accurate estimate of the "true" population mean.
• There is a theoretical distribution of all possible random samples of a given size. It has the standard normal shape with standard deviations that correspond to the same respective areas under the curve as the standard normal distribution you have learned about already!
• The theoretical distribution of all possible random samples of a given size has a standard deviation that is almost never known! You can estimate it by computing the standard deviation of your sample and dividing by the square root of your sample size! This standard deviation has a special name=standard error!

DESCRIPTIVE VS. INFERENTIAL STATISTICS

Introduction to inferential statistics

In the 1991 hit movie "What About Bob?", psychiatric patient Bob Wiley tells his therapist something like, "There are only two kinds of people in the world, those who love Neil Diamond and those who don't."

Some might call Bob's statement a "false dichotomy", but there really are only two kinds of statistics: Those that DESCRIBE and those the INFER. 

 

So far, we have been dealing with those that describe. Now, we are going to start talking about statistics that infer! (It is a great day!)

Descriptive statistics are very nice to us...they don't care about anything other than describing the data that we have in front of us. 

The real fun comes when we use statistics to scientifically predict things. When I was a child, I always wanted to become a weather person...I dreamed about how everyone would love me because of my ability to predict the weather. Almost like magic, I would predict impending weather disasters and save entire cities. Well, I became a statistician, and you are becoming one too! So, we can do similar things (and, sadly, sometimes get it just as wrong as the weather people :(  More on this later...)

Samples: Because collecting data from everyone is too costly!

The entire group of people* you want to study is called the "population". If you can get information from everyone in your population, you are all set! There is nothing else to do besides the descriptive statistics we have been learning: mean, median, mode (measures of central tendency), standard deviation, variance, skew, kurtosis, and maybe some nice visuals: bar chart, pie chart, histogram, frequency table and so on.

Those things tell you all about--or describe--your population.

So, where do inferential statistics come in? We need inferential statistics when we are no longer able to gather data from the entire population!

Gathering data from every person in the entire population is called a "census". Hence the name of the U.S. Census that happens every 10 years--they are trying to gather data from everyone!

Q. So, why not gather data from the whole population?

A. It is too costly!

If you can afford to collect data from every member of your population (the group of people that you want to study), then do it! Usually, that is not the case.

{Quiz}

If you need to know how many people in your office prefer cheese pizza, pepperoni, or black olive, then what is your population?

{Quiz}

Thanks for taking that quiz! Hopefully by now you realize that your population is the group of people you want to study!

After you ask everyone in your office, you might have a frequency table like this:

	Fx	%
Cheese	5	35.7%
Pepperoni	7	50.0%
Black Olive	2	14.3%
TOTAL	14	100.0%

Now, there is nothing else to do! You know you will need 35.7% of your pizza to be cheese, 50% to be pepperoni and so on (hopefully the pizza place is good at math!).

But what if your population is on a bigger scope? Sociologists, doctors and others are often interested in national trends (or even global).

So, what if your question is: "What are the pizza preferences OF AMERICANS?"

Imagine that. Really imagine that. If your next assignment said, "Find out the pizza preference of each and every American." How would you do it? A survey would cost you THOUSANDS of dollars even if you had already identified every American. Even then, perhaps only 10% will respond without an incentive. Maybe if you gave everyone a $25 gift card for participating, you could get 50% of people to respond. That would cost you OVER $4 BILLION! Let's make it a $5 gift card and assume we still get a 50% response rate (which we probably won't). That is only (*sarcasm alert*) going to cost you $750,000,000. All of this isn't even taking into account the time it will take to do the surveys and analyze the data or what it will cost to hire staff to do all of those things.

Bottom line: We need a better way!

It turns out that we can estimate (key word alert!) ESTIMATE these preferences by taking a sample, a smaller number of people from our total population.

YOU WILL NOW BE LEARNING 2 RULES OF SAMPLING IN DETAIL:

1) If you use a random sample, you can use it to estimate the "true" statistics of your population (The "true" statistics of your population are called parameters. So we have Sample Statistics and Population Parameters! SS, PP).

2) The larger your sample is, the more confident you are that it is a good reflection of the population parameters.

Now, allow me to translate into English from Statistics language using the pizza example:

When you cannot get information about pizza preferences for every single person in the group you are interested in studying, you can use a smaller number of people to estimate pizza preferences for the large group! Using this smaller group means that we may or may not get numbers that match the large group. But we can improve our chances of it by doing two things. First, choose your smaller group without any "rhyme or reason". More specifically, you need to use a method of selection that gives every person in the population a chance to be chosen, and that gives everyone an equal change of being chosen. Rolling dice is a good example (as long as it is not a weighted di from Las Vegas!). The second thing is the get the biggest number of people that you can! The more people you have, the more likely it will be that your numbers match what is really going on in the population!

Whew! See why Statistics is a helpful language? Instead of the whole paragraph above, once you know Statistics language we can just say: You may use a random sample to estimate population parameters and those estimates will be less biased the larger the sample is.

Don't worry! You will get the hang of all of this as we go...The key is to remember that Samples have Statistics and Populations have Parameters.

*Here, we will use the term "people" but it is not limited to that. If you are studying Redwood Trees in a certain forest, your population is the group of trees. It might also be fish, amoebas or anything else that you want to study!

FREQUENCY TABLES: PART II

Hopefully FREQUENCY TABLES: PART I is permanently emblazoned in your mind and heart. If not, here is the recap of the major points:

POINT #1: Frequency tables are all about summarizing COUNTS or the frequency with which something occurs, BUT NOT ALL NUMBERS IN A FREQUENCY TABLE REFER TO COUNTS! **Be sure you take the time to differentiate between numbers that represent COUNTS or FREQUENCIES and other numbers.

POINT #2: The column on the left is a list of VALUES that someone in the dataset provided. (They are NOT counts even if they are numbers).

POINT #3: FOCUS FIRST ON THE COUNT! Whatever you are doing with the frequency table, make sure you first recognize which column refers to the counts, and which columns do not. This is especially essential if the VALUES are also numerical. 

POINT#4: Interval/ratio variables are terrible candidates for frequency tables! This is especially true when they are continuous variables. (If you need a refresher on levels of measurement, click here). However, people can and commonly do make frequency tables by changing your variable (for example, to ordinal or nominal variables). 

POINT #5: Counts can tell you where the mode is, but counts are NEVER the mode. Students, repeat: COUNTS ARE NEVER THE MODE. They just tell you which category is the most frequent response, but the category itself is the mode. 

The mean and median can also be uncovered through frequency tables, but we have to expand them a little first. 

STEP 1 (VITAL!!): Determine your level of measurement. You may remember that NOMINAL VARIABLES DO NOT HAVE A MEAN OR A MEDIAN. 

	add, subtract, multiply, divide… (MEAN)	Greater than/less than (MEDIAN)	Difference (MODE)
Ratio	X	X	X
Interval	X (w/ caution)	X	X
Ordinal	-	X	X
Nominal	-	-	X

Because the mean requires addition and division, it only applies to ratio and interval variables. The median is the middle number when the values are arranged from lest to greatest, so it is only possible to compute it for ratio, interval and ordinal variable because nominal variables cannot be arranged from least to greatest. Mode applies to all levels of measurement because it is simply the most common response.

STEP 2: If you have a nominal variable, make the frequency table as shown in part I, find the most common category (the mode) and you are done!

If not, expand your table.

STEP 3: The first step in expanding the table is to find the cumulative frequency.

CUMULATIVE FREQUENCY=the total count up to a given point. Here is an example that builds on the GPA frequency table from PART I.

Picture #1

So, Column B hold the counts for each value, and Column C (our new column) holds the counts up to a certain point. Here it is crystal clear:

VERY LOW: There are 0 people in this category and it is the first category so it is everything up to that point.
VERY LOW through LOW: There are 3 people from "Very low" through "Low". From the beginning through the "Low" category, there are only 3 people (0 in "very low" plus 3 in "low").
VERY LOW through HIGH: There are 5 people from "very low" through "High". From the beginning through the "High" category, there are 5 people (0 in "very low" plus 3 in "low" plus 2 in "high").
VERY LOW through VERY HIGH: There are 10 people from "very low" through "very high". From the beginning through the "very high" category, there are 10 people (0 in "very low" plus 3 in "low" plus 2 in "high" plus 5 in "very high").

STEP 4: Add a new column, "cumulative percent". It is just like it sounds--the percent of people that the cumulative frequency represents.

Picture #2

The calculations appear in Column D, Cumulative Percent, but usually only the percentage appears. The calculation is a courtesy to make it more clear. Because there are 10 total observations (0+3+5+2=10, or just look at the biggest/last number in the Cumulative Frequency column) we divide each number in the Cumulative Frequency column (Column C) by the total (in this case 10).

This means that from the beginning through "very low" we have 0% of the total observations. From the beginning through "low" we have 30% of the total observations and so on. This is how we can find the median.

The median occurs at the 50% mark. (We call these "percent marks" percentiles). What is the median in Picture #2?

REMEMBER, THE MODE IS NEVER THE COUNT! THE COUNT TELLS YOU WHERE THE MODE IS, BUT IT IS NEVER THE MODE!

You don't have to pass it like this though...

Also, the 50% mark (50th percentile) after all the "HIGH" responses are accumulated, so we need to pass it! We do not PASS the 50% mark until the VERY HIGH category. BUT, the same rules apply here as in the past and because we have an even dataset, the median can technically be between two different values. Arrange them in order like you are familiar with and you will see:

LOW, LOW, LOW, HIGH, HIGH, VERY HIGH, VERY HIGH, VERY HIGH, VERY HIGH, VERY HIGH.

So our median here is between HIGH and VERY HIGH. And that's exactly how you report it with ordinal data: Between HIGH and VERY HIGH.

STORE IN LONG-TERM MEMORY: The same even numbered dataset rule applies to the median as before. If you see the exact 50th percentile in your frequency table, the median will be between the corresponding value and the next value. If it is an odd numbered set, you will not see the exact 50th percentile, and the median is in the category corresponding to the point where you have passed the 50th percentile.

Picture #3

Here is an odd numbered, similar version of the last dataset. What would the median be here?

Now the median is firmly within the "HIGH" category because that is the point where we have PASSED the 50th percentile.

STEP 5: Calculating the mean. Remember this figure of a frequency table of a ratio-level continuous variable (GPA)?

Picture #4

These are the kinds of frequency tables that are not very helpful. They are also the only kind for which you could calculate the mean. It is good practice to learn how to do it, and it may appear ON THE TEST in your stats class.

Let's go back to the "Slices of Pizza Eaten" frequency table from the last unit.

Picture #5

Here is the expanded table that also shows us how to compute the mean:

Picture #6

Again, focus on the COUNT! Look at the 1 slice of pizza row. The count is 4. So 4 people said that they ate one slice of pizza. Similarly, 1 person said they ate 2, 1 said they ate 3, 2 said they ate 4 and 2 said they ate 5. So we really have this:

1 1 1 1 2 3 4 4 5 5

You can see how you could simply add this to get: 1+1+1+1+2+3+4+4+5+5=27.

However, you could also make it easier by doing 4(1)+2+3+2(4)+2(5)=4+2+3+8+10=27.

Basically, the frequency table does the second. We simply multiple the value of each response by the count for that response. Then, we add up all of those multiplied values and divide by the total. Remember, the biggest value in the cumulative frequency column is the total number of observations. (Here the red arrow is pointing it out).


MAJOR TAKEAWAYS:

• Frequency tables can be expanded to include new columns: cumulative frequency (the total frequency up to a given point), and cumulative percent. (You could also add a category percentages column just be dividing the frequency of each response/category by the total number of observations--not discussed here).
• The level of measurement of the variable being used in the table determines the measures of central tendency that you can compute:
• Ratio/interval=mean, median, mode
• Ordinal=median, mode
• Nominal=mode only (discussed only in PART I)
• The median is the CATEGORY (not the COUNT) that corresponds to the point where you pass the 50th percent mark (percentile). If there is an exact 50th percent mark in the table, the media is right between that and the next category. 
• As in PART I, KEEP YOUR EYE ON THE COUNT!

FREQUENCY TABLES: PART I

A frequency table is a simple thing. Here is one:

PICTURE #1

...but BE CAREFUL it can be deceptively simple! It has trapped and deceived many statisticians over the years. You will be its next victim--or would have been except for this mini lesson. In just a few minutes, you will be Master of the Frequency Table.

A few keys to avoid deception:

POINT #1: Frequency tables are all about summarizing COUNTS or the frequency with which something occurs, BUT NOT ALL NUMBERS IN A FREQUENCY TABLE REFER TO COUNTS! **Be sure you take the time to differentiate between numbers that represent COUNTS or FREQUENCIES and other numbers.

Look at PICTURE #1 again. Which column of numbers are counts and which are not?

COUNT RECOGNITION

1. WHICH COLUMN IN PICTURE #1 ARE COUNTS?

The left
The right

ONLY the column on the RIGHT represents counts. So what are the numbers in the column on the left?

LEFT COLUMN CHECK

1. What do the numbers on the left in Picture #1 represent?

The number of times each person ate pizza
A total number of slices that was eaten by at least one person in the dataset
The number of people that ate pizza
A value of all theoretically possible numbers of slices that might be eaten by people
None of these

See how it can get tricky?

The column on the left is a number of pieces that at least one person ate. The column on the right is the NUMBER OF PEOPLE that ate that many pieces. MORE ON THIS LATER. For now just let it incubate as you move on to the next point.

POINT #2: The column on the left is a list of VALUES that someone in the dataset provided. It is clearer if we use values that are not numbers. Watch:

Picture #2

Now there is little (to no) confusion about which column contains the counts (frequencies) and which contains the values of the responses.

• 4 people said "bus" is their preferred mode of transportation
• 1 said "car (driving myself)"
• 1 said "car (getting a ride)"
• 2 said "skateboard"
• 3 said "bike"
• 5 said "walk" 
• and 1 said "other"

Easy to see. Now use the same eyes to look again at Picture #1:

Picture #1

How many people answered with each of the possible responses?

FIND THE COUNTS:

1. How many people answered with each of the possible responses?

1 person ate 4 slices, 2 ate 1, 3 ate 1, 4 ate 2 and 5 ate 2
4 people ate 1 slice, 1 ate 2, 1 ate 3, 2 ate 4 and 2 ate 5
1 slice ate 4 people, 2 slices ate 1 person, 3 slices ate 1 person, 4 ate 2 and 5 ate 2
None of these is correct
All are correct, in a sick way...

Not this count...

Starting to see it now? If not, here is tip #3 to master frequency tables:

POINT #3: FOCUS FIRST ON THE COUNT!

You simply will NOT fail if you focus first on the counts. In the first row of Picture #1, where is the COUNT? It is the column on the right. This is all a little facetious because it is labelled "Count". Don't worry, this is your chance to master the skill of "focusing on the COUNT". Surprisingly, this column often will be labeled either as "counts" or "frequency" or just and "f". Your job, first and foremost is to find that column. It is the absolute ground zero of the frequency table. That is why it is called a frequency table.

NOW YOU TRY>

Picture #3

Make a frequency table for the GPA data by filling in the worksheet below. Your grade will appear on the right.



If you figured that one out--A+ to you! If not, you are probably in really good company at this point. Why? Because frequency tables can be terribly tricky and deceptive as simple as they appear to be.

REMEMBER, FOCUS ON THE COUNT!

Not this count...

And the COUNT lives in Column B! Column A is just telling us all the values that were actual responses. And in this case, *THERE IS NO DUPLICATE VALUE* You notice this if you are focused on the count because you would have noticed that there is only 1 total COUNT of any given response!

In other words, everyone has a different GPA in this case, so there is one response for each value and each value gets its own row:

Here is the answer key. If you didn't get it, no sweat! This was a tough one--as long as you learned to focus on the COUNT, you came out with what you needed to learn!

Picture #4

This brings up the next point to avoid being tricked by frequency tables:

POINT#4: Interval/ratio variables are terrible candidates for frequency tables! This is especially true when they are continuous variables. (If you need a refresher on levels of measurement, click here). However, people can and commonly do make frequency tables by changing your variable. For example, what if we made this more of an ordinal variable by having just 4 categories instead of the exact GPA?

• 0-0.9999 (Let's call it "VERY LOW GPA")
• 1.0-1.9999 (Let's call it "LOW GPA")
• 2.0-2.9999 (Let's call it "HIGH GPA")
• 3.0-3.9999 (Let's call it "VERY HIGH GPA")

(Notice that the ".9999" endings make it so there is no overlap between categories. Otherwise even numbers would be included in both. E.g. 2.0 would be included in both "1.0-2.0" and "2.0-3.0" categories.)

Now let's look at the new table next to the old table (the colors illustrate the way we combined the categories):

Picture #5

Notice how we start to notice some trends now. For example, a lot of people have "very high" GPAs and no one has a "very low" GPA (grade inflation at work!). you may also notice that it tells a story about measures of central tendency (MEAN, MEDIAN and MODE). Let's quickly revisit those terms in a way that is so simple it almost doesn't do justice to them:

MEAN: All the numbers added up, divided by the total number of observations.
MEDIAN: The middle number when they are all sorted from smallest to largest (or the average of the two middle numbers if it there is an even number of observations).
MODE: The most common number.

You can find all three from this table! The MODE is the CATEGORY (>AHEM< THE "***!!!CATEGORY!!!***" )with the biggest COUNT. Keep your eye on the COUNT!

In this case, what is the mode in our new table with the new categories?

2
3
5
None of these is correct

KEEP YOUR EYE ON THE COUNT! If you answered 2 or 3 or 5, you were WRONG!! Why? Because THOSE ARE ALL COUNTS! Is the median GPA at a school the COUNT of some category? If you ask me the median GPA at Frantuckanilly State University and I told you 2,342 (the COUNT of people with an average GPA) would it make an sense to you?

POINT #5: A count (frequency) is NEVER the mean, median or mode!!

Read that sentence again 1,822 times. A count (frequency) is NEVER the mean, median or mode!

The count tells us where the mode is, but it is NOT the mode. Think of someone telling you the average GPA at their university is 9,286 and you will get the point.

So, in this case, what is the mode? Hopefully, if you got the mini quiz wrong you answered 5, because 5 is the count that indicates the mode--very high.

The mode is "very high" because it was the most frequent response.

you can also find out the mean and median, but to do this, we need to expand our frequency tables. You will learn how to do that in FREQUENCY TABLES: PART II. 

WHO IS REY?

If you saw Star Wars Episode 7 you may be wondering, Who is Rey? Rey herself said, "I am no one." Bu there is little doubt we will find out more about her past in the coming two movies. However, with release dates deep into the future, you may feel too anxious to wait! The Google search "Who is Rey?" Generates over 120 MILLION hits. The internet is ablaze with conversations and debates about what will be revealed in the next two movies.

If you are lucky enough to have some skills in statistics, you may be able to get ahead of the game. Remember that show "Who Wants to be a Millionaire?"? It was amazing that the poll the audience answers seemed to yield the correct response so often. Perhaps you could use statistics to "poll the audience" and it will give us the correct answer.

The problem is that with 129,000,000 matches and multiple possible theories being expressed on each matching web page, the task is formidable. Random sampling makes it possible to take a smaller number of those webpages and still end up with the same answer--at least it gives us a range that we are somewhat confident about (see other posts on sampling and confidence intervals on this cite).

First, it can be helpful to do some pre-research so that we have some idea what we are looking for. I have done it for us this time around and found nine theories (some related to others) and twelve criteria that people say should be satisfied by the theory.

The theories:

First, have a look at the nine theories:

• The Obi Wan's granddaughter theory (daughter of Luke and Obi Wan's daughter)
• The daughter of Luke Skywalker theory
• The daughter of Luke and a "Mara Jade" type Jedi theory
• The daughter of Luke and a "Mara Jade"-turned-evil theory
• The daughter of Leia and Han Solo theory
• The daughter of Leia-turned-evil and Han Solo theory
• The daughter of Obi Wan Kanobi theory
• The conceived by the force theory
• The reincarnated Anakin Skywalker ("Chosen one") theory

These were gathered by a purposive selection of articles that seemed most relevant. Purposive sampling means that articles are chosen based on the information they can provide. The results of purposive samples are not generalizable (able to be applied to the whole population) but can be an excellent choice in exploratory research or in pre-research because it helps the researcher get their bearing with major themes relevant to the study. In this case, it helped us uncover nine theories that we kept hearing over and over again.

There could be millions of potential theories, but we can know when to stop doing our pre-research when we start hearing only the same major theories over and over again. (We sometimes call this saturation).

The criteria:

Next, there were also twelve criteria that people kept talking about. According to the articles and comments read, these are things that the theory should satisfy for the theory to be chosen by those making the movie. A good theory should:

• Have the ability to explain Rey's advanced abilities with the force
• To explain her advanced pilot skills
• To explain her advanced mechanical skills
• To explain why she was abandoned (or placed) on Jakku as a young child
• To explain Maz Kanata's statement that Rey's family is not coming back
• To explain the draw Rey has to Luke's/Anakin's lightsaber
• Keep the movies Skywalker family-centric (to meet a statement made by creators of the movies)
• Create an interesting plot twist
• Be true to the Star Wars feel (and perhaps parts of even the expanded universe EU)
• Explain Obi Wan's "first steps" line in Rey's vision
• Explain cinematographic allusions or foreshadowing about who Rey is (like the look of her clothing or her upbringing on a dust planet).
• Explain the apparent connection between Rey and Leia toward the end of the movie

There are two ways to look at the "worthiness" of a theory vis-a-vis the criteria: How well a theory meets each criterion, and how many of the criteria it meets. One theory may provide a fascinating and excellent explanation about Rey's ability as a pilot but completely fail to explain Rey hearing Obi Wan's voice during her vision. Similarly, one theory may provide a fair explanation of all these criteria, but another may provide phenomenal explanations for half of the criteria. 

This will have to be sorted out later. But, just keep in mind that our evaluation of these theories will be some combination of how good the theory is at explaining each criterion, and how many of the criteria the theory explains. These could be called "depth" and "breadth" respectively. 

Preliminary results

There are many possible ways to go about this but one is to make a crosstabulation (crosstab). This simply means that we put the categories of one thing as column headers, and the categories of the other as row headers, and they we will in the frequencies we observe at the intersection. (In practice we usually make two variables and the "intersect" or "cross" them using statistics software). 

For now, I have filled in each cell with my subjective analysis based on my readings. 

The table is below:

This is simply the product of me rating each theory (in rows ->) as + , ++ , or +++ where + means "the theory would provide a decent explanation" and +++ means "the theory would provide a very good explanation". Notice there is also a - rating, meaning "this would provide a bad explanation of the criterion".

Now, back to the two ways of assessing the results: depth and breadth. If you scroll over to the right, you can see the total number of points each theory got. This is its overall strength and is simply the total number of + that it has, minus the total number of -. So + gets 1 point, ++ gets 2 points, +++ gets 3 points and - gets -1 point.

To the right of that is another column that give each theory a point for each criterion that it satisfies with at least one + .

Results

Rank by total points (depth):

1. Reincarnated Anakin/Chosen one
2. Daughter of Luke and "Mara Jade"-turned-evil
3. Daughter of Han Solo and Leia-turned-evil
4. Daughter of Luke and "Mara Jade"
5. Luke's daughter
6. Force conceived
7. Daughter of Luke and the daughter of Obi Wan's
8. Daughter of Han Solo and Leia
9. Daughter of Obi Wan

As we see here, the more simple versions of theories are less able to provide strong explanations for the different criteria on average. The top theories involve more details and usually, a woman that has turned evil. The internet world tends to find some appeal in the idea that there will be a "Rey, I am your mother" moment over the next two movies at some point. The idea is that either "Mara Jade"/Luke's Jedi wife turned evil in a sort of Darth Sidius reversal. She feels that the only way to conquer the dark side is to make one's way into it and then destroy it from within. Luke, disagreeing with this philosophy parts ways with his wife and has to hide their young daughter (Rey) and wipe her memory. Nevertheless, she has some Jedi training from Luke (and possibly ghost Obi Wan) that resurfaces later on, making Rey as powerful as we see in the movie. So, Luke's estranged wife will turn out to be either Snoke (a disguise) or Phasma, and, upon learning of Rey, try to bring her to the dark side. Some in this camp even think it may be Ben's intention to do the same, thus the scene where he looks at the Darth Vadar mask and says, "I will finish what you started"--not referring to destroying all Jedi, but to restoring balance to the force by passing through the dark side. 

A similar vibe runs through the Leia-turned-evil theory--that she is not pleased with the inability of the republic to put down the first order and decides to take a small band of resistance fighters to do it. Thus, the resistance is portrayed as a small movement with rudimentary spacecraft rather than more elaborate ships and equipment. This could play out with some similar "Rey, I am your mother" moments, and many believe that Leia might also be behind Snoke (as a disguise). Other possibilities behind Leia's turn to the dark side might be related to her inability to face the darkside like Luke did with Darth Vadar, her lack of training in the force by the light side, or, possibly the same thing that would turn a "Mara Jade" character to the dark side--fighting it from within. 

The #1 theory, however, does not have this tone. Instead, it portrays Rey as a reincarnation of Anakin, or the Chosen one. In this theory, the "Chosen one" is not a single person, but takes on many different personas over time through reincarnation. Thus, Rey is, in a sense, Anakin, out to undo the mistakes of his past life. We see Rey on a dusty run-down planet, good at flying and fixing things, and some argue that Rey bears a remarkable resemblance to Shmi Skywalker. This theory obviously explains a lot of criteria because it is sort of the catch-all--instead of answering how she is related to Anakin (a point that seems to be pretty overtly made by the movie makers), it simply asserts that she is him. However, this theory provides great depth of explanations for a lot of the criteria, but not as much breadth as others. For example, it does not offer a ready explanation of why she was abandoned as a young child or hears Obi Wan Kenobi's voice in her vision. 

Let us look at the ranking by breadth--percent of criteria satisfied:

1. Daughter of now-evil "Mara Jade" and Luke
2. (tie for 1st) Daughter of Han and now-evil Leia
3. Reincarnated chosen one 
4. Daughter of "Mara Jade" and Luke
5. (tie for 4th) Luke's daughter
6. Daughter of Obi Wan's daughter and Luke
7. (tie for 6th) Obi Wan's daughter
8. Daughter of Leia and Han
9. Force conceived

Once again, there is the general trend that more complex theories involving women turned evil are at the top! In fact, the same theories occupy the top three places, but the "reincarnated chosen one"theory has fallen to 3rd. This is because, while it satisfies many of the criteria very well, it does not satisfy as many of the criteria as some other theories.

Conclusion

Short of drawing up and conducting a full survey to a representative sample of the Star Wars fan universe, we conclude that three of the theories seem to land at the top:

1. Daughter of now-evil "Mara Jade" and Luke
2. Daughter of Han and now-evil Leia
3. Reincarnated chosen one 

So, which should be crowned the best? Here, we might want to weight depth or or breadth differently--implying that one is more important than the other. But, assuming we weight them equally, we just average them and end up with these final standings:

1. Daughter of now-evil "Mara Jade" and Luke
2. Daughter of Han and now-evil Leia
3. (tie for 2nd) Reincarnated chosen one 

So, in episode 8 when most of the world finds out that Rey is the daughter of Luke and a "Mara Jade" type Jedi-turned Phasma who was trained by Luke and ghost Obi Wan but had her memory wiped and was sent to Jakku to protect her from the dark side and her mother, maybe you will be able to say you heard it here first thanks to the power of analysis.