Making decisions in the face of variability TWSSP Wednesday
Agenda for today • Larger Sample Size • Standard Deviation • Raisins • Memorizing Words • Gummy Bears • GOALS: • Understand how a distribution changes when more data points are sampled • Understand how standard deviation appears in a graph • Understand what factors can influence differences in two groups • Know when a difference between two groups is statistically significant • Understand one technique for testing for statistical significance
Standard Deviation • Measure and record your hand span in mm • Make a dot plot of the class data and use a wedge (▲) to mark the mean below the horizontal axis. • I notice, I wonder • What are two sources for the variability in the data?
Standard Deviation • Make a second dot plot where the data points are the differences (deviations) between each hand span measurement and the mean • Use a wedge to mark the mean of the deviations • I notice, I wonder • Could you get from the first plot to the second without any computations? • How would you describe the ‘typical’ deviation from the hand span mean?
Standard Deviation Calculate the deviations from the mean Draw horizontal lines off of the mean to show the deviations
Standard Deviation • Draw a line the length of which is the average of the deviations from the mean. Don’t worry about whether the deviation is positive or negative. Estimate the length of the line. • This length is a good estimate for the standard deviation, a measure of the extent to which data varies from the mean, without consideration for whether a data point is above the mean or below.
Estimate the standard deviation • Mean is 2.5
Standard Deviation • For each of the pairs of graphs on your handout, decide which has the greater standard deviation, and why.
Raisins • How do you know when the store brand is a better buy than the name brand? • Price is one factor to consider, but quantity is important, too!
Raisins • Plot the data for the quantity of raisins in small boxes of both generic and name-brand in a way that enables you to draw conclusions about the relative quantities of each. • I notice, I wonder • What features of the plot should you consider in your decision of which raisins to buy?
Raisins • These data are for small boxes of raisins. What would you predict about quantities in large boxes of raisins? Why?
Memorizing Words • You will receive a list of 20 words. You have 1 minute to memorize as many words as possible. • Swap with a partner and count number correct. • Analyze the data, considering both the numerical representation and a graphical representation.
Memorizing Words • Research study • When researchers conducted this study, they found that the difference between the mean of the meaningful words memorized and the mean of the nonsense words memorized was 4.27 • What might have caused this result?
Memorizing Words • Assume treatment has no effect. Is it still possible that the difference in the means could occur? • Is it probable that the difference in means could occur in the absence of a treatment effect? • How might we determine if the difference in means is more likely a result of the treatment or a result of variability?
Memorizing Words • Simulation • Write the following data set on index cards so one data point is on each card • 12, 15, 12, 12, 10, 3, 7, 11, 9, 14, 9, 10, 9, 5, 13 • 4, 6, 6, 5, 7, 5, 4, 7, 9, 10, 4, 8, 7, 3, 2 • Shuffle the cards well and deal them into two groups of 15. • One is Set A, the other is Set B • Find the mean score for each group, and calculate the difference in means (mean A – mean B) • Repeat the simulation 20 times
Memorizing Words • Based on the results of the simulations, what conclusion would you be more likely to draw and why? • There is no real difference between the treatments, and the researchers were unlucky in the random assignment to have found the difference of 4.27 • I am now convince that there is a genuine effect from memorizing meaningful words instead of nonsense words • If the mean for the meaningful group were only 2 words larger than that for the nonsense group, what would you conclude? Why?
Memorizing Words • There are two types of variability to which we must attend in this task. What are they? • When it can be determined that it is more likely for an effect to be the result of a treatment than the result of randomness, the results are considered to be statistically significant.
Memorizing Words • You have the graphs for 3 other classes to compare with the original data. • For each case, do the results provide more convincing evidence or less that the meaningful words list was easier to memorize? Explain. • Make up 15 hypothetical results for the meaningful words group and 15 for the nonsensical words group that would help you to be very convinced that the meaningful words list made a difference. What factors are you considering in making your lists?
Gummy Bears • Will gummy bears travel farther if they are launched from a steeper height or a lower height? • 3 sets of 10 launches, find the mean of each set of 10 • Create a representation of your data to show in a gallery walk. • After the gallery walk, do you think height of launch pad matters? • Organize the mean data • I notice, I wonder
Gummy Bears • Comparing the data • How can you summarize the difference in distances between the two conditions? • Is there variability in the measurements for one condition? How do you see that variability? What do you think causes it? • Is there variability between the two conditions? How do we see it and what causes it? • What, if anything, could you do to minimize the variability?
Gummy Bears • Based on these data, are you willing to say that the height from which a gummy bear is launched causes it to travel farther? • What are important parts of an experiment which are necessary to show causation?
Gummy Bears - Analysis • Determine if the difference in travel distance means is statistically significant • Is it caused by the launch height, or is it random? • Devise and implement a plan to test for statistical significance. • Consider what is feasible to do in the next 30 minutes • Consider what is necessary to test for significance • Prepare to report the results of your tests.
Exit Ticket (sort of…) • What would you expect to happen to variability when a sample size is increased? • In your own words, explain what standard deviation measures and what it tells you about a set of data. • When you are comparing two data sets, what features of their graphs should you consider? How does each feature help you to compare the data sets? • What are the possible sources of variability between two data sets? Why do we need to consider the different sources?