DECK 4: UNIT 1 REVIEW MIX Flashcards
1
Q
P(Z>0)
A
The probability that a randomly chosen z score is to the right (above) Zero (the middle) is the same as the percent of area above a z score of zero in a normal model.. which is about 50% or 0.5
2
Q
What does SHIFT and SCALE mean?
A
Shift is when you add or subtract, scale is when you multiply
3
Q
What is variability?
A
Differences. how things differ. There is variability everywhere.. We all look different, act different, have different preferences. Statisticians look at these differences.
4
Q
What is the mean?
A
the old average we used to calculate. It is the balancing point of the histogram
5
Q
What is Q2 also known as?
A
the median
6
Q
not associated is the same as being ____________
A
independent
7
Q
Compare DATA-STATISTIC-PARAMETER using categorical example
A
Data are individual measures like meal preference: taco, taco, pasta, taco, burger, burger, taco. Statistics and Parameters are summaries. A statistic would be 42% of sample preferred tacos and a parameter would be 42% of population preferred tacos.
8
Q
What percent of the data is above Q3?
A
25%
9
Q
If you want to calculate the probability (%) something falls between two values in a normal model, what do you do?
A
find z scores for both value, and then normcdf (Z LOW, Z HIGH )
10
Q
Year in school (F,S,J,S) and Pizza Preference (pepperoni or cheese) are __________ because _______________
A
independent because all grades have similar preference distributions.. 40% cheese, 30%pepperoni, 20% veggie 10% other
11
Q
What does normcdf do?
A
It gives you the area under the normal curve between any two z scores
12
Q
What is the total area under the normal curve?
A
1 or 1.000
13
Q
What is the IQR?
A
Interquartile range… a measure of spread. Q3-Q1. The distance from Q1 to Q3. The regular range is Hi-Lo, this is the inner range, the interquartile range.
14
Q
When there is a relationship between two variables, we say that they are
A
associated (or not independent)
15
Q
what happens if you multiply all of a data set by a constant? Think of an example
A
it is scaled Both center and spread are impacted. Mean/ median/ stand dev/ iqr/ quartiles all multiplied by that constant. Center, spread and all individual values are changed. Consider 1,2,3,4,5 mean of 3 and range of 4. Now multiply by 10: 10,20,30,40,50 and you get a mean of 30 and a range of 40… both multiplied by ten.
16
Q
What are DESCRIPTIVE STATS?
A
Make a picture. Describe to me the data that you collected, use pictures or summaries like mean, median, range.
17
Q
are there any normal samples?
A
no, nothing is normal, just normalish. The only normal thing is the model we use.
18
Q
How can you turn OGIVES into histograms?
A
RECTANGLE DROP! (bin drop)
19
Q
Compare population to sample
A
populations are generally large, and samples are small subsets of these population. We take samples to make inferences about populations. We use statistics to estimate parameters.
20
Q
How do students often mix up IQR and St. Dev
A
They INCORRECTLY think that Q1 is 1sd below the mean and Q3 is 1sd above the mean. THIS IS NOT TRUE!!! Q1 is only .67 sd above the mean and Q2 is .67 below
21
Q
What is the difference between quantitative and categorical variables?
A
Quantitative variables are numerical measures, like height and IQ. Categorical are categories, like eye color and music preference
22
Q
For information purposes, which gives LEAST… stem-leaf, histogram or box-whisker?
A
Box/Whisker, BE CAREFUL. you really don’t know how things are distributed. The box and whisker and fish tank give a very GENERAL look.
23
Q
Make a guess as to what relative cumulative frequency is?
A
It is the ADDED up PERCENTAGES.. An example is selling candy, 25 pieces sold overall…, with 10 the first hour, 5 the second, 3 the third, and 7 the fourth hour, we’d take the cumulative frequencies, 10, 15, 18 and 25 and divide by the total giving cumulative percentages… .40, .60, .64, and 1.00. Relative cumulative frequencies always end at 100 percent. OGIVE
24
Q
How can you describe spread?
A
range, IQR, stand dev, variance, or simply say: From here, to about here
25
How do you describe SPREAD for unimodal and symmetric distributions?
use the standard deviation
26
If I take a random sample 20 hamburgers from FIVE GUYS and count the number of pickles on a bunch of them? and the average number of pickles was 9.5, then 9.5 is considered a \_\_\_\_\_\_\_?
statistic. (it is a summary of a sample.)
27
When drawing a normal model, what are the PERCENTILES from left to right?
2.5, 16, 50, 84, 97.5
28
What is the difference between discrete and continuous variables?
Discrete can be counted, like "number of cars sold" they are generally integers (you wouldn't sell 9.3 cars), while continuous would be something like weight of a mouse: 4.344 oz.
29
which calculator function gives you a percent?
normcdf(Z left, Z right)
30
When there is no relationship between two variables, we say they are
independent (or not associated)
31
What is a categorical variable?
Qualitative variables are like categories: Blonde, Listens to Hip Hop, Female, yes, no, etc.
32
Think of the minimum value, the median and the IQR, which is impacted by shifting (adding a constant?)
adding a value shifts the entire histogram to the right, so the min and the median will increase by that amount, BUT THE IQR WILL NOT CHANGE.
33
How do you match OGIVES to histograms?
RECTANGLE DROP!!
34
If the distribution is bimodal or multimodal, what would you use for center and spread statistics?
Talk about each mode (center) and maybe use the range or IQR. You could also say "one group seems to go from __ to __ and the other from about __ to \_\_"
35
How do you find 5 number summary from OGIVE?
Split the y axis into quarters. Shoot out to the right from 0, .25, .50, .75 and 1.00 till you hit the line in the ogive, then go straight down. Those numbers on the x axis below correspond to the 5 numbers.
36
Does the IQR capture 68% of the data?
NO. it catches the middle 50%.
37
How do you describe SPREAD for bimodal or multimodal?
talk about the outer edges of the clusters "from here to here" or use the IQR.
38
marginal distribution
distribution in the margins (outside of the table). The overall distributions of a single variable in contingency table.
39
What is the five number summary?
min, Q1 , Q2(median), Q3 and max
40
What percentile is Q3?
75th
41
What are INFERENTIAL STATS?
Look at your data, and use that to say stuff about the BIG PICTURE. like tasting soup. a little sample can tell you a lot about the big pot of soup (the population)
42
How can you match boxplots to histograms?
USE THE FISH TANK METHOD!
43
What is data?
Any collected information. Generally each little measurement. Like, if it is a survey about liking porridge, the data might be: yes, yes, no, yes, yes. if it is the number of saltines someone can eat in 30 seconds, the data might be 3, 1, 2, 1, 4,3 , 3, 4
44
Can numbers be CATEGORICAL?
sure. Zip codes, sports jersey numbers, telephone numbers, social security nunmbers, area codes… these are categorical.
45
What is a Z score?
The number of standard deviaiton away from the mean
46
What is meant by cumulative frequency?
ADD up the frequencies as you go. Suppose you are selling 25 pieces of candy. You sell 10 the first hour, 5 the second, 3 the third and 7 in the last hour, the cumulative frequency would be 10, 15, 18, 25
47
Give a simple example showing that adding a constant doesn't change the spread, but changes the center. (this always happens)
Data set: 1,2,3,4,5 Spread (range):4, Center: 3 add three and get new data set: 3,4,5,6,7 spread:4 Center: 5 (center went up, spread stayed the same). The IQR and SD will stay the same, but median and mean go up 3. Called shifting, or sliding the data.
48
When drawing a graph or chart, what do you have to remember to do?
LABEL AXES, make a KEY(if needed ) AND GIVE IT A NAME!!! "Figure 1: Age and Food Preference"
49
Does a census make sense?
A census is ok for small populations (like Mr. Nystrom's students) but impossible if you want to survey “all wild turkeys on Nantucket”
50
What percent of the data is between Q1 and Q3?
50%
51
How do you find percentiles and make a boxplot from OGIVE?
Go across till you hit the curve and then STRAIGHT DOWN!
52
What is Statistics?
The study of variability
53
How do you describe SPREAD for skewed distributions (or distributions with outliers?)
Use the IQR
54
What are 2 branches of AP STATS?
Inferential and Descriptive
55
Compare data to parameters
Data is each little bit of information collected from the subjects?. They are the INDIVIDUAL little things we collect? we summarize them by, for example, finding the mean of a group of data. If it is a sample, then we call that mean a "statistic" if we have data from each member of population, then that mean is called a "parameter"
56
Why do we plug 999 into normcdf?
It needs a z score, but we can't plug in infinity. So we go down or up 999 standard deviations and that pretty much gets everything
57
what happens if you ADD a constant to each value in a data set?
it is SHIFTED only. Does not impact spread. This effects all of the data values and measures of center (mean, med) and quartiles, deciles, etc, IT DOES NOT CHANGE THE SPREAD! (IQR, St Dev, Range all stay the SAME).
58
where are the "outlier fences?"
1.5 IQR above Q3 and 1.5 IQR below Q1. Just a rule of thumb.
59
Think of the minimum value, the mean and the standard deviation, what is impacted by shifting (adding a constant)
adding a value shifts the entire histogram to the right, so the min and the mean will increase by that amount, BUT THE SD WILL NOT CHANGE.
60
How do you find Q1 and Q3?
Q1 is the median of the bottom half and Q3 is the median of the upper half (they are the 25th and 75th percentiles)
61
What is a population?
the group you're interested in. Sometimes it's big, like "all teenagers in the US" other times it is small, like "all AP Stats students in my school"
62
what is the emperical rule?
mean 68-95-99.7 yeah!
63
If the distribution is unimodal and symmetric, what would you use for center and spread statistics?
Mean (center) and Standard Deviation (spread)
64
How do you describe CENTER for unimodal and symmetric distributions?
use the MEAN
65
what does P(z\< -1) =?
The probability that a randomly chosen z score is to the left (below) -1 is the same as The percent of area below a z score of -1 in a normal model.. which is about 16%
66
What is a statistic?
A numerical summary of a sample. Like a mean, median, range or SD of a sample.
67
Gender and Video Game playing are\_\_\_\_\_\_\_\_\_\_\_ because\_\_\_\_\_\_\_
associated (or not independent) because a higher percentage of males play video games. (think.. It depends on gender)
68
what is the shortcut normcdf?
gives % from raw data,
skips Z score.
normcdf (low VALUE, high VALUE, mean, sd)
69
Give an example of independent variables
If 80% prefer cheese and only 20% prefer pepperoni IN EACH GRADE AT MONOMOY…then they all have the same preference, so grade doesn't matter. We say "school year and pizza choice are independent"
70
We are curious about the average wait time at a Dunkin Donuts drive through in your neighborhood. You randomly sample cars one afternoon and find the average wait time is 3.2 minutes. What is the parameter?
The parameter is the true average wait time at that Dunkin Donuts. This is a number you don't have and will never know. The statistic is "3.2 minutes." It is the average of the data you collected. The parameter of interest is the same thing as the population parameter. In this case, it is the true average wait time of all cars. The data is the wait time of each individual car, so that would be like "3.8 min, 2.2 min, .8 min, 3 min". You take that data and find the average, that average is called a "statistic," and you use that to make an inference about the true parameter.
71
are any populations actually normal?
no, nothing is normal, just normalish. The only normal thing is the model we use.
72
What is a census?
Like a sample of the entire population, you get information from every toad in the valley
73
What is quantitative data?
The actual numbers gathered from each subject. 211 pounds. 67 beats per minute.
74
If the mean is above the median, the distribution may be
skewed right... the mean follows the tail
75
What is meant by relative frequency?
The PERCENT of time something comes up (frequency/total)
76
mean/SD/median/IQR. How do I know which ones to use?
when unimodal and symmetric, mean and sd. If skewed or outliers. Median and IQR. If bimodal? Talk about the MODES
77
data or datum?
datum is singular.. Like "hey dude, come see this datum I got from this rat!" data is the plural.. "hey look at all that data Edgar got from those chipmunks over there!!"
78
What percent of the data is below Q2?
50%
79
associated is the same as \_\_\_\_\_\_\_\_\_\_
not independent
80
not independent is the same as
associated
81
What percentile is Q1?
25th
82
the output for normcdf (Zleft, Zright) is\_\_\_\_\_\_\_
the area under the normal curve between the given z scores
83
what is the shortcut invnorm?
gives data value from percentile,
skips Z score.
Invnorm (percentile, mean, sd)
84
What does invnorm do?
It gives you the Z SCORE from a percentile
85
Association and Independence. How are they related?
Variables are either independent or associated. Meaning: if one impacts the other then we say there is an association. If not, Then they are independent.
86
What does GSOCS stand for?
Gaps Shape Outliers Center Spread (put on your gsocs when comparing distributions) be sure to talk about each one clearly (make a list)
87
If you are tasting soup.. Then the flavor of each individual thing in the spoon is the \_\_\_\_\_\_\_\_, the entire spoon is a \_\_\_\_\_\_.. The flavor of all of that stuff together is like the _____ and
If you are tasting soup. Then the flavor of each individual thing in the spoon is DATA, the entire spoon is a SAMPLE. The flavor of all of that stuff together is like the STATISTIC, and you use that to MAKE AN INFERENCE about the flavor of the entire pot of soup, which would be the PARAMETER. Notice you are interested in the parameter to begin with... that is why you took a sample.
88
When do we often use mode?
With categorical variables. or bimodal things…. For instance, to describe the average teenagers preference, we often speak of what most students chose, which is the mode. It is also tells the number of bumps in a histogram for quantitative data (unimodal, bimodal, etc?). or if there are TWO HUMPS we use the modes to describe the center.
89
independent is the same as \_\_\_\_\_\_\_\_\_\_
not associated
90
What is a sample?
A subset of a population, often taken to make inferences about the population. We calculate statistics from samples.
91
How can you tell if variables in a contingency table are independent?
If the distributions are the same across the variables.. Then it doesn't DEPEND… so INDEPENDENT. Ex: 30% of freshman and 30% of seniors like cabbage.
92
How do you find relative frequency?
just divide frequency by TOTAL.
93
If you want to find % below a value, what do put into normcdf (? ?)
find z score for value, and then normcdf (-999, Zright)
94
How can you describe the center of a distribution? (how can you think about the different summaries)
OPTIONS: give the mean (balance), median (splits area in half), mode (peaks, if bimodal talk about both modes) or say "centered around \_\_\_\_"
95
What do OGIVES look like?
They all start at the bottom left (0%) and go to top right (100%)
96
What symbols do we use for population standard deviation and sample standard deviation?
Sigma for population and s for sample.
97
What is a "percentile?"
It tells you the percent of data BELOW a certain value
98
Q: How can you get a parameter?
A: By taking a \_\_\_\_\_\_\_\_\_\_\_
Census
99
What is a CUMULATIVE FREQUENCY GRAPH?
An OGIVE. It shows the added up totals as you go left to right.
100
How do you describe distributions (histograms)?
Shape-Cener-Spread- and STRANGE (Outliers and gaps)
OR… some say GSOCS. where's yo GSOCS?
(gaps, shape, outliers, center, spread)
101
If the distribution is skewed (or outliers/not symmetric) what would you use for center and spread statistics?
Median (center) and IQR (spread)
102
How do you find a certain percentile on an OGIVE?
Start at the % on the Y axis.. travel horizontally to the right until you hit the line, then straight down to the X axis. That data value is the percentile.
103
What are random variables?
If you randomly choose dogs from a list, then their hair color, height, weight and any other data collected from them can be considered random variables.
104
Which is more sensitive to outliers and skewed? Mean, median. SD or IQR?
Mean and SD are most influenced by outliers. median and IQR are RESISTANT, RESILIENT, ROBUST!!
105
Compare Descriptive and Inferential STATS
Descriptive tells you about the data that you have, inference uses that data you have to try to say something about an entire population.
106
What is the difference between a bar chart and a histogram
bar charts are for categorical data (bars don't touch) and histograms are for quantitative data (bars touch)
107
What symbols do we use for population proportion (%) and sample proportion (%)?
p for population and p-hat for sample
108
When are box plots used most often?
When comparing a bunch of different sets of data.
109
Which calculator function gives you a z score?
invnorm(%ile)
110
If a distribution is skewed right, what will be greater, the mean or median? WHY?
Mean. The mean moves further to the right to keep balance.
111
If you want to calculate % above a value, what do you put into normcdf(? ?)
find z score for value, and then normcdf (Z left, 999)
112
P (Z \< 1 ) ?
The probability that a randomly chosen z score is to the left (below) 1 is the same as The percent of area below a z score of 1 in a normal model.. which is about 84%
113
what is a conditional distribution?
A distribution with a condition (within the table), along only one row or one column… NOT IN THE MARGINS. You are given a condition.. Then read along that row or column.
114
What is the median?
the middlest number, it splits area in half (always in the POSITION (n+1)/2 )
115
Use the following words in one sentence: population, parameter, census, sample, data, statistics, inference, population of interest.
I was curious about a population parameter, but a census was too costly so I decided to choose a sample, collect some data, calculate a statistic and use that statistic to make an inference about the population parameter (aka the parameter of interest).
116
What is a standard deviation?
average (typical) distance to the mean (about). It is how far you expect a random value to be away from the middle.
117
What is a contingency table?
shows distributions across 2 variables like gender and music pref. AKA 2-way table
118
Give a quick example of associated variables
A higher percentage of boys play video games than girls so we say "gender and video game playing are associated" or "gender and video game playing are not independent"
119
What is the variance?
The average squared distance to the mean. Or the SD2 (It is the SD before you take the square root, so it is the stuff under the radical in the formula)
120
How many SD wide is the IQR in a normal distribution?
NOT 2!!!! Think about it. The middle 68% is 2 sd wide, since the IQR is only the middlest 50% it must be less than 2. try [invnorm(.75)] x2. You find that it is only 1.35 SD wide if the distribution is nearly normal.
121
What is the difference between quantitative and categorical data?
The data is the actual gathered measurements. So, if it is eye color, then the data would look like this "blue, brown, brown, brown, blue, green, blue, brown? etc." The data from categorical variables are usually words, often it is simply "YES, YES, YES, NO, YES, NO" If it was weight, then the data would be quantitative like "125, 155, 223, 178, 222, etc.." The data from quantitative variables are numbers.
122
What is a frequency distribution?
A table, or a chart, that shows how often certain values or categories occur in a data set.
123
How can you think about the mean and median to remember the difference when looking at a histogram?
mean is balancing point of histogram, median splits the area of the histogram in half. MODE IS THE HIGHEST POINT
124
P (-2 \< Z \< 2)
The probability that a randomly chosen z score is between -2 and 2 is the same as The percent of area between -2 and 2 in the model, which is 95%
125
If I take a random sample of 20 hamburgers from FIVE GUYS and count the number of pickles on a bunch of them? and one of them had 9 pickles, then the number 9 from that burger would be called a\_\_
a datum, or a data value.
126
What percent of the data is above Q3?
25%
127
How do you describe CENTER for bimodal or multimodal?
talk about the modes (the lumps, the clusters)
128
What do we sometimes call a categorical variable?
qualitative
129
What symbols do we use for population mean and sample mean? (and sample sd and population sd?
Mu for population mean, xbar for sample mean. (sigma for pop sd and just an s for sample sd)
130
What percent of the data is below the median?
50%
131
Think of the minimum value, the median and the IQR, which is impacted by multiplying by 10?
If you multiply a data set by 10, then the min, median and the IQR will multiply by ten
132
What are the percentiles for Q1, med, and Q3?
25, 50 and 75
133
Compare data to statistics
Data is each little bit of information collected from the subjects. They are the INDIVIDUAL little things we collect. We summarize them by, for example, finding the mean of a group of data. If it is a sample, then we call that mean a "statistic" if we have data from each member of population, then that mean is called a "parameter"
134
How do you describe CENTER for skewed or distributions with outliers?
use the MEDIAN
135
Another name for "skewed right" is
positively skewed
136
what is a clear example of the medians resilience and when you would use the median instead of the mean?
(change just the top value). Imagine if we asked eight people how much money they had in their wallet. We found they had {1, 2, 2, 5, 5, 8, 8, 9}. The mean of this set is 5, and the median is also 5. You might say "the average person in this group had 5 bucks." But imagine the same group the next week, but one of them just got back from the casino and the dist was (1, 2, 2, 5, 5, 8, 8, 9000}, in this case, the median would still be 5, but the mean goes up to over 1000. Which number better describes the amount of money the average person in the group this time? 5 bucks or 1000 bucks? I think 5 is a better description of the average person in this group and the 9000 is simply an outlier.
137
What is the mode?
the most common, or the peaks of a histogram. We often use mode with categorical data
138
What percentile is the median (aka Q2)?
50th
139
What is the difference between categorical VARIABLES and categorical DATA?
The Variable is the overall category. Like "EYE COLOR". The data is the actual measurement from the subjects. Like "blue, brown, blue"
140
Why don't we always use the mean, we've been calculating it all of our life ?
It is not RESILIENT, it is impacted by skewness and outliers
141
What percent of the data is between Q1 and Q3?
the middle 50%. That is the IQR
142
When can you round?
AT THE VERY END!!! (keep at least 3 digits until end!)
143
What is the difference between a sample and a census?
With a sample, you get information from a small part of the population. In a census, you get info from the entire population. You can get a parameter from a census, but only a statistic from a sample.
144
How do you find the median from an OGIVE?
go halfway up the y axis, then shoot across to the curve, then straight down. It's at the 50th percentile (halfway up)
145
Compare DATA-STATISTIC-PARAMETER using quantitative example
Data are individual measures, like how long a person can hold their breath: ?45 sec, 64 sec, 32 sec, 68 sec. That is the raw data. Statistics and parameters are summaries like the average breath holding time in the sample was 52.4 seconds and a parameter would be the average breath holding time in the population was 52.4 seconds
146
How can you describe shape?
TWO THINGS: modes and symmetry.
unimodal, bimodal, multimodal or uniform,
symmetric, skewed
147
What is a quantitative variable?
Quantitative variables are numeric like: Height, age, number of cars sold, SAT score
148
What is a parameter?
A numerical summary of a population. Like a mean, median, range, or st. dev of a population. Those are all parameters.
149
How does multiplying by a constant impact the summary statistics of a data set? (or random variable)
It is SCALED. Both center and spread are effected. They all (mean, median, IQR, SD, range) get multiplied by three. (BE CAREFUL, remember the variance is the SD squared, so the variance gets multiplied by 9).
150
What is frequency?
How often something comes up
151
Think of the minimum value, the mean and the standard deviation… what happens if you multiply a data set by 10
If you multiply a data set by a ten, then the min, mean and the SD will multiply by that number.
152
If asked to compare distributions, what should you write about?
A sentence comparing the SHAPES. A sentence comparing the CENTERS. A center comparing the SPREADS. and a sentence comparing the STRANGE STUFF. (GSOCS)
153
If you want to find percentile for a value, what do you put into normcdf (? ?)
find z score for value, and then normcdf (-999, Zright) like going from negative infinity up to the z score
154
How are mean, median and mode positioned in a skewed left histogram?
goes in that order, mean median mode
155
Why are there different standard deviation formulas for population and sample? Are they the same thing?
Both equations are actually doing the same thing. They both attempt to calculate the true population proportion. When you have all of the data from the population you just divide by n and get the actual SD. BUT If you only have a sample then you are using that to make a guess (inference) at what the population standard deviation is.. What happens is that samples tend to have less spread so their SD underestimates the population, BUT, when you divide by n-1 instead of n, It gives you a better estimate of what the population standard deviation is.
156
If I take a random sample of 20 hamburgers from FIVE GUYS and count the number of pickles on a bunch of them? and I do this because I want to know the true average number of pickles on a burger.. what is that called?
a parameter, a one number summary of the population. The truth. AKA the parameter of interest.
157
What is categorical data?
The actual individual category from a subject, like "blue" or "female" or "sophomore"
158
When we say the average teenager are we talking about the mean?
It depends, if we are talking height, it might be the mean, if we are talking about parental income, we'd probably use the median (because it is skewed), if we were talking about music preference, we'd probably use the mode to talk about the average teenager.
159
What symbols do we use for population mean and sample mean?
Mu for population mean, xbar for sample mean. (sigma for pop SD and s for sample SD)
160
What do you call things that are not independent?
associated. Or not independent. We generally don't say DEPENDENT (unless talking about y variable on a scatterplot).
161
What is the mode?
the peaks of a histogram (the humps). or with categorical data, the most popular category
162
What is the difference between a population mean and a sample mean?
population mean is the mean of a population, it is a parameter, sample mean is a mean of a sample, so it is a statistic. We use sample statistics to make inferences about population parameters.
163
What is a random sample?
When you choose a sample by rolling dice, choosing names from a hat, or other REAL RANDOMLY generated sample. Humans can't really do this well without the help of a calculator, cards, dice, or slips of paper.
164
What is the difference between a parameter and a statistic?
BOTH ARE A SINGLE NUMBER SUMMARIZING A LARGER GROUP OF NUMBERS. But pppp parameters come from pppp populations and sss statistics come from ssss samples.
