The Idea: There are many different ways to gather a sample…

The easiest sampling plan is just the simple random sample. That's like putting everyone's name from the population into a hat and drawing your sample from that.

An Example: You want to know the opinions that the 500 bowling league participants have about the snack bar facilities of a bowling alley. To get a sample of 100 of these league players, you take a list with all 500 names and draw 100 of them at random.

Sometimes you know there are specific groups that might differ in their results and so, in a stratified sampling plan, you take separate random samples within each group.

An Example: You are conducting a poll about possible solutions to the problem of sexual assaults on campus and you know that men and women are likely to have different opinions. In a stratified sampling plan you would take a separate poll of women and men and then put those two samples together.

In a cluster sampling plan, you sample naturally occurring groups and then pick everyone within the sampled groups for your final sample.

An Example: You want a sample of students who are in one of 100 classrooms. A cluster sample might pick 10 classrooms and then question every student in those 10 classrooms.

In a systematic sample you take, say, every tenth person on a list into your sample.

An Example: You want to know what mobile carrier is being used by people waiting in line to get into a store to buy a new type of cellphone, so you interview every 5th person in the line.

Finally, there are often disastrous sampling schemes that do not allow you to gauge the probability that particular elements of the population are sampled. One type of non-probability sampling is a volunteer sample where the sample chooses itself by volunteering to participate.

An Example: You want to know how well the audience of a radio station likes the music of a new band so you advertise a toll-free number and interview anyone that calls in.

Another disastrous scheme is a convenience sampling plan where you just sample the people easiest to find and interview.

An Example: You quickly gather audience opinion about a new play by talking to everyone in the front row of an auditorium because they are the easiest to reach and talk to.

Online notes: Section 2.4

The Idea: Reliability deals with how consistent a measurement is (ask yourself: if you repeat the process over and over will the numbers I get be very close to each other or not?). Bias deals with whether the measurement is on target (ask yourself: if you take lots of measurements will they average around the true value?). Validity deals with the appropriateness of the measurement (ask yourself: does this get at the concept I am trying to measure?)

An Example: I pace off a room to measure how wide it is and come up with 15 paces. Then I state that "this room is 15 yards wide" (i.e. I am using my paces to measure yards)

• Reliability issues:
  • I am not a robot so my paces are pretty inconsistent. One time I might get 15 paces and another time I might get 13 paces or 16 paces for that same room. That is the hallmark of an unreliable measurement
• Bias issues:
  • I am a short guy so my paces are around 31 inches long on average (rather than 36 inches for a yard). Thus, every time I use paces and call them yards my numbers are too high (more of my little paces than there are yards in the lengths I'm trying to measure). Also, I can't walk a perfect straight line so again I will get too many paces when I try to measure a length. These are two ways that my paces are biased
• Validity issues:
  • paces do get at the concept of "length" pretty well so they are fairly valid in that sense. But if you really want the concept of "length in yards" then the reasons given about bias would be an indication of a lack of validity

Online notes: Section 1.3

The Idea: The five-number summary uses the quartiles to try to draw out the salient features of the distribution of a measurement variable. Thus, it separates the data into compartments that hold about 25% of the data each. Its values are plotted in the boxplot.

An Example: For 2013 model year cars, the EPA reports that the worst gas mileage (combining city and highway driving) was for the high-performance Bugatti Veyron that got just 10 miles per gallon (mpg). The best gas mileage was for the Toyota Prius C that got 50 mpg that year. We can see how those extremes compared to the rest of the distribution by noting that the 25th percentile amongst all 2013 model year cars was \(Q_L = 16\) mpg, the median mileage was 19 mpg, and the 75th percentile was at \(Q_U = 22\) mpg. Of course the values in the upper tail of the distribution were more spread out (the top 25% went from 22 mpg out to 50 mpg) than the values at the low end (the bottom 25% went from 10 mpg to 16 mpg) due to the very high mileage achieved by the hybrids. This indicated a somewhat skewed to the right distribution.

Online notes: Section 3.4

The Idea: The mean is the numerical average while the median is the 50th percentile (half the numbers are bigger and half are smaller). The median is a better descriptor of the concept of “a typical value” while the mean gets at concepts related to a total. In a skewed histogram the mean is drawn toward the tail with the outliers while the median is unaffected by outliers.

An Example: The average household income in Pennsylvania in 2013 was \$70,685 while the median household income that year was \$52,548. The median income is a better reflection of the income of a typical household while the mean income is a better reflection of the total income flow in the state. The mean is higher because the distribution of household incomes is skewed to the right with a few outliers on the high end.

Online notes: Section 3.3

The Idea: Examining the variability in a measurement is a core statistical issue. The standard deviation measures how far data values come from the mean. If there was no variability, every measurement would be the same – all being the mean value - and the standard deviation would be zero. As a rule of thumb, about 68% of the values come in a symmetric histogram come within one standard deviation of the mean. However, the standard deviation is quite sensitive to outliers so it does not provide a good measure of variation for describing a heavily skewed distribution. A resistant measure of variation is the Inter-Quartile Range (IQR). It gives the range of the middle 50% of the data by looking at how far you need to go to get from the 25th percentile to the 75th percentile.

An Example: A study of risk factors associated with low infant birth weight was conducted at Baystate Medical Center in Springfield Massachusetts and found a mean birth weight of 2945 grams (that’s about 6 pounds 8 ounces: a triangle marks the position of the mean in the histogram below) with a standard deviation of 729 grams (about 26 ounces). The IQR was 3478 – 2413 = 1065 grams. A picture of the histogram and boxplot show how these measures of variability compare. In this case about 71% (shaded area in histogram) of the babies weighed within one standard deviation of the average so the rule of thumb worked pretty well.

Online notes: Section 3.5

The Idea: The distribution of a variable tells us the proportion of time each possible value occurs and, for a measurement variable, the distribution is displayed in a histogram. This includes skewed to the right where higher values are more spread out and most values are bunched up on the low side; skewed to the left where lower values are more spread out and most values are bunched up on the high side; symmetric where the distribution shows a mirror image on both sides of the middle; and bimodal where there are two prominent peaks in the shape.

An Example:

Online notes: Section 3.2

The Idea: Standard scores (or z-scores) give a measure of relative standing on a list by telling you how many standard deviations above (+) or below (-) the mean you are. To interpret a standard score, first look at the sign; negative z-scores indicate a value below the mean while positive z-scores indicate a value above the mean. Next, look at the size of the number. Values within one standard deviation of the mean (z-scores between -1 and +1) are pretty common and will usually make up most of the distribution. Values outside of three standard deviations away from the mean are quite rare and indicate an extreme value either on the low side (z-scores less than -3) or extreme on the high side (z-scores bigger than +3).

An Example: NBA MVP Stephen Curry of the Golden State Warriors is an excellent free throw shooter, making about 91% of his free throws in the 2014-2015 season. That year, the league average for starting players was 78% with a standard deviation of 9%. Curry’s standard score was then \(z = (91 – 78)/9 ≈ 1.4\). You can see from his standard score that his free throwing shooting was a good deal above average. However, the distribution of free throw percentage accuracy is a skewed left distribution with a few very poor shooters being extreme outliers on the low side (e.g. DeAndre Jordan shot only 40% from the free throw line for a z-score around -4). That inflates the standard deviation and makes the interpretation of standard scores a bit more difficult. Remember, z-scores work best for symmetric histograms.

The Idea: Percentiles tell us the number that has a specific percentage of the values in a distribution below it. The 90th percentile has 90% of the values below it and 10% above; the 60th percentile has 60% of the values below it and 40% above, etc.

An Example: NBA superstar LeBron James formally of the Cleveland Cavaliers made only 71% of his free throws during the 2014-2015 season. This put him at the 16th percentile amongst starting players. While he excelled at many things, 84% of starting players had a greater free throw shooting accuracy.

Online notes: Section 4.1

The Idea: When the histogram is symmetric (and especially if it is close to the bell-shaped normal curve) then the empirical rule gives us a good idea of how the standard scores are related to where the data are located. About 68% will have standard scores between -1 and +1 and about 95% will have standard scores between -2 and +2. If the number of standard deviations away from the mean is not a round number, we can look up the z-score on the internet to get percentages.

An Example: Subjects in a study of a new diabetes treatment had an average systolic blood pressure of 130 with a standard deviation of 15 and their blood pressures approximately followed the normal curve.

Online notes: Section 4.2