More on Probability Models and Expected Values

Today was a wrap up of probability models and expected value. We were in a computer lab today, so I spent some time on using Minitab as students begin to gather and analyze data for their projects. I was hoping to get into normal models today as well but that will have to wait until next week.

I used roulette as an example to have students create probability models and calculate expected values. I like using this since the probability models are simple and it turns out that all of the expected values are the same, regardless of whether you be on a specific number, bet on even or odd, bet on red or black, bet on the 2nd 12 numbers, bet on the center column; every expected value turns out to be the same. This does two things: 1) it provides easy practice for students to create a probability model and calculate the expected value and 2) makes students understand that a game like roulette is designed based on probabilities and expected values. The expectation from the casino's perspective is that you will lose. This is how casinos can build ostentatious hotels and provide customers free rooms and free drinks.

After, we took a look at a specific probability model, the binomial model. This model is generated from Bernoulli trials, which we defined. A Bernoulli trial has three characteristics:

1. There are fixed probabilities.
2. There are only two possible outcomes for each trial.
3. Each trial is independent.

For a binomial model, we are examining the number of successful outcomes for Bernoulli trials for any given amount of trials. For example, the probability of tossing 7 heads in 10 coin tosses.

After examining a binomial model situation we went into how to calculate the probabilities. With the other non-topic items that were being covered today that was as much as we accomplished.

Below is an outline of today's lesson with italicized comments enclosed in square brackets [like this].

·         Roulette expected values [had a student pick a number and then used this to calculate different expected values based on the color and number picked, i.e. red, even, 1-18, etc.]

·         Coin flipping questions

   o   Have students calculate probabilities

   o   What are characteristics of these problems [posed this question and students came up with all three requirements.]

      §  Two possible outcomes

      §  Constant probability

      §  Independent trials

      §  These are Bernoulli trials

      §  Number of successes in fixed trials is binomial

   o   How did you go about calculating these probabilities?

      §  Success/failures and number of ways

      §  Give binomial formula and explain

      §  Discuss how to calculate with a calculator [Used a baseball player getting 4 at bats and possessing a .300 batting average. What is the probability of getting exactly 2 hits and what is the probability of getting at least 3 hits.]

   o   What is the mean and standard deviation for a binomial model [Asked class how many heads they would expect to toss in 10 coin tosses. Discussed how the obtained 5 and then went into formulas. Showed the standard deviation only to reinforce that a probability model possesses both a mean and a standard deviation.]

      §  np and sqrt(npq)

Bayes Theorem, Probability Models, and Expected Value

Based on where things were left off, I thought a little more work with Bayes Theorem was warranted. But first, we had to finish discussing the Birthday problem. Most of the class hadn't made any headway on the problem, although a couple had calculated the probabilities for up to seven people in the room.

I went through the calculations, which resulted in

    P(no one with the same birthday in a room of 30 people) = 364! / (365²⁹ x 336!)

It's a nice formula but most calculators and spreadsheets get an overflow error trying to calculate values this large. I demonstrated how lists could be made, manipulated and used to calculate the result. It works out to approximately a 70% probability. This is surprisingly large; most people think the value would be much smaller. This lies at the heart of statistical analysis; should we be shocked by something we observe or is it really something that is expected. In this case, with such a high probability, it is not shocking that at least two people in a room of 30 have the same birthday.

From there, we worked with another Bayes problem involving jumping paper frogs. The frogs are indistinguishable but one frog lands on its feet more often than the other. We pick a frog and it lands on its feet. What is the probability that we picked the one that lands on its feet the most often. Most students did a good job creating a tree diagram for the problem, and with minor prompting most were able to determine the desired "reversed" probability.

I then asked what the probability would be if the frog we selected landed on its feet twice in a row. This led to some discussion as to whether or not the probability would change; some thought it would be the same. After calculating the probabilities, they saw that the probability grew larger that the better jumper was selected. This makes sense; every time we see another feet-first landing our confidence we feel more confident that we picked the better jumper.

After, we looked at probability models. I used simple examples like rolling a single die or flipping a coin. I then asked students to consider what the average of roles would be if we rolled a single die numerous times. They were unclear, so I simulated 100 die rolls and calculated a sample mean of 3.58. I then asked how many times we should expect to roll a 1 if we tossed a die 1200 times, and how many 2's etc. After calculating the total point values and dividing by the number of rolls, we came to a result of 3.5. I repeated the question but as what the average would be if we rolled 12,000 times and 12,000,000 times. The expectation is that the average should be 3.5. This is not a value you can roll but what we expect the average of all our rolls to become.

We worked through a problem involving sibling counts and the percentage occurrence of each. I asked students to calculate the expected number of siblings. One student suggested multiplying the value times the percentage. Referring back to the die roll, the class saw that this worked. I then used this to define expected value. We practiced on a simple coin tossing game: two heads wins $1, one head and one tail wins $0.50, and two tails loses $1.

Below is the outline of what was covered today along with italicized comments enclosed in square brackets [like this].

o   Bayes

§  Jumping frogs

o   The Birthday problem

§  Results involve factorials and permutations 

—Discrete Random Variables

·         What is a random variable?

o   Variable that depends on chance [discussed concept briefly]

·         What is a discrete random variable?

o   Can list out all possible values of variable [used die roll as an example]

·         When make a table and histogram of a variable it is the probability distribution and probability histogram

o   Use dice roll example [made histogram on calculator after simulating; with thousands of rolls would expect histogram to flatten out showing a uniform distribution]

·         What is the mean dice roll?

o   This is called the expected value [expected value and mean of the probability model are equivalent terms]

o   Use sibling example to calculate the expected value

o   What does this value mean?

·         What is the standard deviation of a dice roll?

o   Consider how standard deviation was calculated before [showed how to calculate on a calculator and then showed formula]

o   What does this value mean? [as with previous standard deviation, the average distance from the mean]

Conditional Probability and Independence

The ideas of conditional probability and independence were covered today. Using a table, it's relatively straight-forward to demonstrate what conditional probability is. I tried a new demonstration using card guessing and hints to help illustrate independence. Ultimately, I used the definition of P(A | B) = P(A) for independence, then went into the general multiplication rule P(A) x P(B) = P(A | B) x P(B) to show that an equivalent demonstration of independence would be P(A) x P(B) = P(A and B).

These ideas are the easy part. Going into Bayes Theorem is much more confusing. I like to use tree diagrams and work through determining the probability that we are interested in and then focusing on the probability that is of interest out of that total amount. I tell students to not get hung up on the formula but to consider what we are interested in calculating. The tree diagrams help to sort this out and, as probabilities are calculated for the different branches, we ultimately perform the calculations used in Bayes Theorem without getting hung up on what pieces fits where within the formula.

I wrapped up class with the birthday problem. This illustrates how quickly a probability calculation can become messy and for students to think about calculating the compliment of the situation.

Below is an outline of today's lesson along with italicized comments enclosed in square brackets, [like this]

·         Conditional Probability and Independence

o   M and Ms

   §  P(orange| peanut) [referenced back to the original M and M problem to kick things off]

o   Asthma and smoking

   §  Table

   §  Venn Diagram [did not demonstrate this today, will reference it tomorrow]

   §  Tree [used the Three Strings tree diagram to illustrate this]

o   Card probabilities

   §  Ask for volunteer

   §  Choose card from deck; student can win $1 if guesses drawn card (no suit necessary); student writes guess on slip of paper which is not shared with anyone

   §  Ask class “what is the probability of having a correct guess?” answer 4/52

   §  For a penny give a hint—card is red (or black)

   §  “what is the probability of a correct guess?” answer 2/26

   §  For another penny another hint—card is a heart (or appropriate suit)

   §  “what is the probability of a correct guess?” answer 1/13

   §  For another penny give one more hint—card is a number (or face card)

   §  “what is the probability of having a correct guess?” answer 1/9 or ¼

   §  Discuss what information was helpful and what wasn’t

   §  Talk about independence informally [this went as described, really had students think about what information influenced any change in guess]

o   Independence

   §  Go back through problems to check on independence

      ·         M and Ms [asked if being orange and type were independent, had students demonstrate using both definitions]

      ·         Roulette [didn't use]

      ·         Asthma [students looked at whether living with a smoker and having asthma were independent]

      ·         Clothing and gender [didn't use]

o   Bayes

   §  P(plain | orange) [students calculated and then showed tree diagram]

   §  Asthma [didn't use, looked at medical testing results problem from book]

   §  Clothing (shorts or jeans, etc depending on what you see in class) and gender [didn't use, may do this next class]

·         Counting and probability
o   The Birthday problem

   §  Have students record day of birth in month column [turned out we had four pairs of same birthday, one of which had three individuals]

   §  Pose question [the question being is it unusual to have a matched birthday or is it something we should expect]

   §  Results involve factorials and permutations [after students worked on the problem a bit, discussed the direct calculation, building from two students to three to four; then talked about calculating no matching birthdays, where the result is much easier to calculate; this was homework, along with working  through some Bayes Theorem problems]

Investigating Probability

The focus today was on looking at probability. I looked at basic probability properties and rules for calculation. Many of these were covered last class but I did introduce mutually exclusive events. I also discussed the concept of independence informally. I used roulette to illustrate many of the properties and rules for calculation.

Afterward, I tried an investigation I saw at the NCTM annual conference. You have 3 strings and you are blindly tying ends together. The possible outcomes are to form one large loop, one medium and one small loop, or three small loops. The class actively discussed what the possible outcomes were and how to identify or count the outcomes. As expected, students tried to apply formulas they had learned in the past without considering the problem situation. I used a tree diagram (a new way to represent probability situations) and we worked through the probabilities. The results were counter to the class's intuition as to what result should be most likely. I liked this activity; it illustrated many components of the basic probability properties and rules and helped show how tree diagrams are created and used for calculating probabilities.

Below is the outline of the class along with commentary that is enclosed in square brackets and italicized, [like this].

·         Reference basic probability rules

·         Roulette practice

o   P(red)

o   P(even)

o   P(3^rd 12)

o   P(1-18 or center column) [did not reference general addition rule here, but noted how we accounted for numbers that were both 1-18 and in center column]

o   P(2^nd 12 and 3^rd column)

o   P(black and even)

o   P(black or even) [now referenced general addition rule]

o   P(not 19-36)

o   P(0 or 00)

o   P(0 and 00)

o   P(1^st spin red and 2^nd spin black) [spins are independent so that probabilities are multiplied]

o   Is the event  “8 and black” mutually exclusive

o   Is the event “25 and black” mutually exclusive

·         Strings

o   Take 3 strings

o   Grab and fold in half

o   Swirl to randomize

o   Tie 2 ends together, do it again, tie last pair together

o   What are possible outcomes [as discussed above, outcomes are: 1 loop, 2 loops, or 3 loops]

o   Which do you think most likely to occur, which least likely to occur [class guessed that 2 loops would be most common]

o   Tabulate results and represent [broke student work to have students share thinking on how they were configuring outcomes; needed to do this as some students were getting themselves off track with thoughts like there are 6 ends being matched with each other so 6 x 6 = 36 total outcomes]

     §  What are actual probabilities, show using trees [worked through a branch and then had students attempt to complete other branches, most got stuck]

A First Look at Probability

Today we undertook looking at probability. I usually start probability off by asking students to think about different properties they may have learned about probability over the years. I use a think-pair-share format. This helps get out what students already know about the topic and helps to uncover misconceptions that need to be addressed.

Normally, I have students that know something about probability, even if it is rather rudimentary. Today, virtually  no one knew anything about probability, other than possessing some misconceptions. It was good to learn that there was little or no prior knowledge about probability and also to address some misconceptions, such as if you flip a coin several times in a row and they all are heads that the probability of flipping a tail increases or that the more unlikely an event the greater the probability that it occurs in a large number of trials.

I covered some basic ideas and used a bowl full of M&Ms with 50 plain and 100 peanut M&Ms as an example. The class was told that 12% of the plain and 15% of the peanut M&Ms were orange. With this information I asked several different questions about the bowl and the probability of their occurrence. I also used the class characteristics of male and female versus those wearing shorts and those not wearing shorts.

Next class will be focused on specific probability rules, representing probabilities, and practicing their use.

Below is the outline of today's lesson, with comments enclosed in square brackets and italicized [like this].

Probability Concepts

·         What probability rules do you know?

      o   Think, pair, share

      o   Run through basic concepts

        §  0 ≤ P(A) ≤1

        §  P(A) = 0 means never happens

        §  P(A) = 1 means always happens

·         How can you represent probability situations

      o   Tables [used coin toss to illustrate a table]

      o   Venn diagrams [used gender and wearing/not wearing shorts to illustrate]

      o   Use M&M problem

·         Calculating probabilities

      o   P(peanut) [discussed how calculated to bring in idea of (desirable outcomes)/(total outcomes)]

      o   P(not plain)

      o   P(orange and plain)

      o   P(orange or plain)

      o   P(orange)

Regression, Influential Points, and Residual Analysis

Today I finished up a whirlwind look at simple linear regression. The class had a lot of questions, especially about R², which is a difficult concept to grasp. As quickly as we are covering this material, I expected there would be a lot of questions.

The first half of the class was spent reviewing what we did and why from the last class. This was coupled with showing students how to calculate regression equations from summary statistics and by using their calculators. I then had the class work through finding regression equations for the different data sets we had and to write a sentence interpreting the b₁ coefficient in context of the problem and a sentence explaining the meaning of the R² value.

After this, I focused on influential points, residual plots, and analyzing residual plots. For our purposes, I defined an influential point as one the substantially affected the slope of the equation. I provided some examples and we discussed that an influential point does not necessarily mean that the data is an outlier in terms of either the response or predictor variable. I used the experience of drawing ovals around a scatter plot to help students see how they could assess potential influential points.

We wrapped up with working with residuals. We looked at different residual plots from the data sets we were working with and discussed whether or not they looked relatively random, which is the main point I wanted them to walk away with.

Below is the outline used for today's class with comments italicized within square brackets, [like this]

Regression and Correlation
o Influential points
o Y-hat = b₀ + b₁x
o Regression equation passes through the mean-mean point so y-bar = b₀ + b₁x-bar
o b₁ = r s_y / s_x [emphasized for those not able to calculate regressions on their calculator]
o R-squared = r² [got into a nice connection between comparing (y - y-bar)² and (y - y-hat)²]

Residual Analysis and Standard Error of Estimate
Assumptions for regression [didn't really get into these specifics]
o Linear relationship [this was emphasized when discussing correlation]
o Errors have
Equal variances
Independence
Normal distribution [since normal distributions have not been covered yet, it didn't make sense to get into this]
o Show plots of residuals
o Have students create a residual plot for orbit drop data

Correlation and Simple Linear Regression

Today felt a little like a whirlwind. The class met in the computer lab today. I started by showing how to access and use basic features of Minitab. After showing how data entry is similar to using an Excel spreadsheet, I gave a quiz on descriptive statistics and comparing distributions. I asked students to enter the different data sets we collected into Minitab when they were done with their quizzes.

After everyone was done with the quiz, I focused on developing some conceptual understanding of correlation, least squares regression line, and R². I used a couple of applets to help and being in the computer lab, everyone was able to try these out.

The first applet was Regression by Eye. I've used this applet many times. I started off having students describe the association: direction, strength, and form. I then focus on trying to quantify the strength of the relationship using values between -1 and 1. A value of -1 indicates a negative slope and all the points fall on the same line. A value of 1 indicates a positive slope with all the points falling on the same line.

I draw an oval around the scatter plot. Does the oval indicate the direction of the relationship. For different scatter plots, do fatter or narrower plots show stronger associations? As students start to consider these, I ask the class to think about the ratio of the oval's width to its length. A perfect positive association would have an oval with no width, suggesting we could look at correlation as expressed by 1 - w/l. A scatter plot with no linear association would have w and l nearly equal. The expression r = 1 - w/l provides a way to estimate correlation. The only piece to add in is the direction, negative associations will have r-values that are negative. Students started using this technique to estimate r and were able to become more accurate with some practice. I told the class that a visual estimate within 0.2 would be acceptable.

Next we focused on the line of best fit. Using the same applet, I had several students draw lines of best fit. Obviously this could continue with the entire class and conceivably we could end up with 30 or more different lines of best fit. One way to consider a "best" line of best fit would be to reduce the amount of error to a minimum. The next applet does just that.

The Least Squares Demonstration applet visually demonstrates the idea of least squares. This applet starts by using the mean line as an estimate. We can always use this line but it is generally not a very good estimator. Moving the line end points we can pass closer to points on the scatter plot. As we do, we can see that the squared error values change for each point. When the sum of these squared error values are the smallest we have a least squares regression line—the "best" line of best fit.

Students played around with this for a little bit and got a sense of how a least squares regression line worked. Returning to the Regression by Eye applet, we could now look at the various lines that we drew and the mean squared error, basically the average of the squared error values. In the sense of least squares, the line with the smallest mean squared error is better than the others. Clicking on the show regression line, we can see how the hand-picked line compares to the least square regression line.

The final conceptual piece to develop was for the R² value. In this case, I use the Least Squares Demonstration applet to help explain R². I have students focus on the response values compared against the mean line. There is variation around the mean line, specifically (x - x-bar) where x-bar is the sample mean. Square these values and sum and you get Σ(x - x-bar)². If we divide by n - 1 and take the square root we would have the sample standard deviation. Unsquared we have the sample variance. I draw brackets on the y-axis to highlight the variation present in the original data, explaining this is the variation that is present in our data.

We can compare are actual values against predictions using the least square regression line. In this case we have Σ(y - y-hat)² where y-hat is the predicted value of y. Re-orient your view to look at how much variation exists around the least square regression line. I draw a bracket around the regression line to show that the variation has narrowed considerably.

How much variation is explained away? On the y-axis, I draw a length equivalent to the variation around the regression line. I then shade out the variation that is no longer present; this is the amount of variation explained away by the regression line. The percentage explained away is the R² value. I then point out that R² = r², where r is the correlation value.

Now that the conceptual piece was covered, we looked at the passing the buck data set. This data had a correlation of .993 and its scatter plot was nearly a straight line. I ran a regression in Minitab and we looked at the equation and the R² value. The equation was time-hat = -1.65 + 0.68 people. We discussed what the values in the equation meant.
.
In this problem, if there were no people in line, it would take -1.65 seconds for the buck to be passed. In the context of this problem that value is not applicable. I explained that this is often the case. The value of 0.68 indicates that one additional person will typically add 0.68 seconds to the time it takes to pass the buck.

We can use the equation to make an estimate, such as how long would we estimate 10 people to take to pass the buck? Substituting values we see that it should take approximately 5.15 seconds.

What about 100 people. We could use the equation but this is so far beyond the range of the sample that it should be viewed with extreme caution. In this case we have extrapolated beyond the scope of the analysis. On the other hand, the rope and knot data would probably hold up well even under extrapolation because of the nature of the relationship between knot and rope length, assuming a taut rope.

With the last few minutes of class, I had students generate summary statistics, scatter plots, and perform a regression analysis with one of the data sets they entered. This enabled students to get a little more comfortable with the software that they will be required to use in their projects.

This was a lot of material to cover in one day, especially given that some of the concepts are difficult to grasp. Next class we'll focus on working through some of the data we generated using calculators. We'll also address relationship of regression slope to correlation and sample standard deviations. Finally, we'll take a look at the residuals to see what they tell us about the quality of the regression equation.

Below is the outline of today's lesson with any comments italicized between square brackets, [like this].

o   How can we measure strength and direction of the association?

   §  Use general direction of the association—positive slope versus negative slope

   §  Look at length versus breadth of association

   §  Use Regression by Eye applet

o   What line best describes this association

   §  Have students draw lines and discuss

   §  How can we determine a best line?

o   Use Least Squares Demonstration applet

   §  Discuss what is happening

§  In what sense is this the best line

o   Use same app to discuss explained variation

   §  What is simplest model you can create?

·         Y = mean

   §  How much variation is explained by regression?

   §  Draw scatterplot and compare variation around mean versus variation around line

   §  Percent accounted for is R²

o   Computer lab

   §  Use drop data [used pass the buck data as I had it copied down already]

·         Estimates and predictions

o   Extrapolation

o   Influential points [will discuss next class]

Comparing Samples and Activities for Collecting Bi-variate Data

We wrapped up uni-variate data exploration. Students shared graphs and discussed their analysis and conclusions. This was a good discussion. I emphasized that any comparison of distributions should discuss the shapes (modes, symmetry/skewness, and outliers/major gaps), center (mean or median), and spread (standard deviation or IQR). I also told students that the conclusion they draw must be supported by the analysis they perform. They will often see data that has no clear-cut answer and they must communicate what they are using to draw a conclusion. This aspect of statistics typically makes students uncomfortable. They are used to coming up with an answer and it is either right or wrong. They'll get over this.

The second half of the class was focused on collecting bi-variate data to use in regression analysis. The class will meet in the computer lab next time, so this was an opportunity to generate data that we can use. We ended up generating 4 different data  sets to work with.

We had time to discuss graphing the data using a scatter plot. I then had students focus on the direction of the relationship (positive or negative), the strength of the relationship (concentrated or dispersed), and the form of the relationship (linear, curved, or shapeless). We'll use these ideas as a foundation for exploring linear correlation.

Below is the outline I followed today, along with italicized comments in square brackets, [like this].

o   Complete river sampling comparison

§  Share and discuss

·         Population versus samples

o   Populations have a mean and standard deviation, we just typically don’t know what they are

o   Designate these using Greek letters mu and sigma

§  When calculating population standard deviation, divide by n

o   Samples are drawn from a population

o   Designate sample means and standard deviations using Latin letters and bar

o   We try to gauge the value of the population mean and standard deviation from the sample data we draw

o   Every sample varies slightly, so we’ll need to understand and account for this sample variation

o   Larger samples should more closely reflect the population values

Day 4—Regression and Correlation

·         Associations

o   Orbit package delivery – part 1

§  Look for association between drop height and distance from target [delivery vehicle is a crumpled piece of paper; target is a coin on the floor; students gather at least 6 drops.]

o   Collect  additional data and use in the computer lab

§  Knotted ropes [how do the number of knots tied in a rope affect its length]

§  Height and arm span (ref. Da Vinci’s Vitruvian man)

§  Pass the Buck [hand-to-hand passing of slip of paper; start with 3 students and continue adding groups of 3 to the end of the line; response variable is time to pass the buck from one end to the other.]

o   Is there an association

§  Scatterplots

§  Direction of association

§  Shape of association

§  Strength of association

o   Work with other data sets as needed [none need, four data sets should be plenty]

Five number summary and box plots

Today's focus was the five number summary and box plots. The class seemed to do okay with finding a data set that had specific means and standard deviations. Most are getting more adroit at using their calculators to assist them with their calculations and analysis.

We then used the data from the two data sets we used last week to calculate mean and standard deviation and focused on finding the five number summary and IQR. This lesson went much as planned. I was hoping to discuss the connections between sampling, estimating using sample statistics, and the population statistics we are seeking to understand. That discussion will have to wait until tomorrow.

Below is the  lesson outline with comments italicized between square brackets, [like this].

o   What is a quartile?

§  Discuss

o   How can we divide a data set into quartiles?

§  Find median then find median of lower and upper halves

§  Use height data on board to illustrate [Used data from two sets that worked on last class for mean and standard deviation.]

o   Range is difference between max and min, we could look at the difference between Q3 and Q1 [Discussed issues with  range, specifically outliers.]

o   IQR = Q3 – Q1, describes spread of middle 50% of data, it’s another measure of spread

    [Asked class to consider what other ways could measure spread. There were a few ideas suggested that were a bit more involved but could work. Students also confirmed that we wanted to make use of 5-number summary data as the basis for the spread measurement. I was pleased that these students were focused on the proper things.]

o   Box plot

§  Create box on a scale (vertical or horizontal)

§  What about whiskers?

§  Define upper and lower limits using Q1 – 1.5IQR and Q3 + 1.5IQR

§  Whisker goes to actual point values within limits

§  All values outside limits are marked with asterisk and denoted as an outlier [There were good clarifying questions at this point and we spent some time on these. The second data set was used as practice. Afterward, I showed the class how the graph could be constructed using their graphing calculators.]

o   You report summary statistics for your sample, either mean and standard deviation or 5# summary and IQR.

§  Mean and standard deviation used for roughly symmetric data

o   Create 5# summaries and box plots for the two data sets

§  Share and discuss [Put both box plots on the same scale. Discussed ways to compare and contrast the graphs. Discussed what the means in drawing conclusions about the  two data sets.]

o   Complete river sampling questions

§  Share and discuss [This piece went  a little longer than planned. Students were grouped into three sets of 10-12 students as they shared data. They had four box plots to create along with summary statistics for the four data sets. They then had to write a few sentences comparing the graphs. I told the class that anyone can see the graphs, the comparison is what they want people to focus on that will justify their conclusion. I'll have students share what they wrote next class and then we'll focus on the issue of always discussing shape, center, and spread.]

Mean and Standard Deviation

Today's introductory statistics class started looking at descriptive statistics. The focus today was on measures of central tendency (mean and median) and the standard deviation as a measure of spread. I have used this similar lesson many times and it helps bring meaning to these three measures.

The class followed the outline closely. The document camera was not working today; someone said there was a power outage last night, which may have affected it. This just meant I had to write things on the board but it didn't really affect the lesson at all.

Below is the outline with some annotations enclosed within square brackets and italicized, [like this].

·         Mean-Median-Mode

o   Have everyone write down height to nearest inch on a sticky note

o   On reverse side, have students write down what they think the average height of the class will be.

o   Have students line up from shortest to tallest

o   Allow students to revise their estimate of the class average now that they see what the heights look like

§  Convey the idea that in statistics, we hypothesize about a variables value, we gather information and use this data to make revisions to our hypothesis

o   Have one student from each end leave line and place sticky note in appropriate bin on board

o   Continue until only one or two students left

o   What do remaining student(s) represent—the median!

o   Look at histogram on board

§  Describe the shape of this graph

§  Where is the mode?

o   Will the mean be larger, smaller or about the same value as the median? [Held this discussion until later.]

o   Calculate the mean [Introduced sigma notation and x-bar. Discussed what the meaning of the mean is. Interestingly, most students knew how to calculate a mean but didn't understand what it represented other than a vague notion of the center. They could not articulate how what this measurement of center represented. It was a good discussion. I didn't have manipulatives but was able to create a mental image of each student holding small disks totaling their height in inches. The task would be to share out the disks until everyone had the exact same number of disks in hand. Taller individuals would be giving up disks and shorter individuals would be taking disks. Eventually, everyone would have the same whole number and a sub-group would possess one extra, the fractional part of the mean value.]

§  How did your personal estimate compare to the actual mean? [Didn't discuss this since I decided to use this as a way to promote the underlying statistical process of estimate, analyze, and revise estimate.]

o   Show graph shapes and ask where mean vs median will lie [First discussed the comparison of median and mean with height data before discussing comparisons. Wrote out the relationships between mean and median underneath graph.]

·         Variation

o   How would you describe the spread of the height data distribution?

o   Range

§  difference between smallest and largest values

§  what are limitations using the range

o   How else might we look at spread?

§  Discuss ideas, focus on those looking at how far away from mean, lead discussion this way if needed

o   What if we take average difference from the mean? [Introduced the term "deviation from the mean" here.]

o   What can we do to account for negative and positive differences cancelling each other out. [One student suggested absolute values--perfect!]

o   Absolute values work but are not computationally convenient, what is another way to manipulate values to eliminate negative values?

§  Remind about distance formula and squaring [No reminder needed. One student suggested multiplying values and all I had to do was ask the class what we could multiply by.]

o   Let’s take squared differences from the mean and average these

o   Put up four values, say 2, 5, 8, and 9. The mean is 6. If you know there are four values with a mean of 6 and you know three of the values (2, 5, and 8) you can find the fourth value. So, when calculating the mean difference from the sample mean, need to divide by n-1 to account for this fact. [I actually just used the values 1, 2, 3, and 4 here.]

o   We started with inches but the mean squared deviations are in squared inches. How do we get back to our original units? [Students said we'd need to take the square root.]

o   Let’s try it on a smaller data set first. Here are two data sets to work with. [At this point I showed students how to use their calculators to calculate these values.]

o   What do the standard deviations tell you about these two data sets?

§  Discuss [Discussion focused on what does the standard deviations tell you about the data sets. To help give additional practice, I asked students to use 10 data values such that the mean would equal 40. For one data set I required that the standard deviation equal 5 (within 0.2) and for a second data set I required that the standard deviation equal 15 (within 0.2). This homework gives students an opportunity to get comfortable working with the formulas and also gaining a better sense of what the standard deviation represents.]

§  Show dot plots of the data [Did not show since document camera was not functioning.]