The first few questions on this worksheet will deal with measures of central tendency. These data types tell us where the center of the data set lies.

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1 Instructions: You are given the following data below these instructions. Your client (Courtney) wants you to statistically analyze the data to help her reach conclusions about how well she is teaching. Answer each question in order from start to finish. Some questions will depend on results from previous questions. To ensure accuracy, you may work in groups. You might also find it helpful to use the your Data Analysis Guide (Chapters of your book!). Report your findings and conclusions to Courtney before you leave class. The Data,,,,,, 56,,, 84, 28,,, 64,, 80,,, 48, 52,, 76, 44,,,, 64,,, 56 The first few questions on this worksheet will deal with measures of central tendency. These data types tell us where the center of the data set lies. 1. Rewrite the data points in ascending order. 2. Using class intervals of length 10, create a histogram of the data. 3. Does the histogram appear to be an approximately normal distribution? 4. For the data set, find the population mean (µ). 5. What would the population mean be if you replaced one of the s with a zero? 6. Use your answer from question (2) to describe the sensitivity of the mean. In other words, how easy is it to change? 7. Write the data above in ascending order. Since there are an even number of data points, find the two middle data points. Take their mean. What you just found is called the median! 1

2 8. How would you have changed the process for finding the median if the data set contained an odd number of data points? 9. Is your median above or below the mean you found above, or is it about the same? Interpret what this means for the above data set. (We haven t talked about this in class, but what you re considering is whether the data is skewed.) 10. Which number(s) occur most often? These are your mode(s). Interpret the meaning of the mode for this data set. Do you think the mode is a useful piece of information in this application? 2

3 In this next section of questions, we will introduce measures of variability. These measures tell us how a data set varies from its center, or its spread. The first few questions will deal directly with quartiles, while the last few questions will deal with variance and standard deviation. 11. For this question, just find the range of the data. Note that the range here is interpreted a little different than the range of a function from algebra. Be careful. 12. Above you had to list the scores in ascending order. Go back to this list and choose the first 15 scores. Find the median of this group of scores. This is the first quartile (q 1 ). 13. Now take the last 15 scores from your ascending list and find the median of this group. This is the third quartile (q 3 ). 14. How does the process for finding q 1 and q 3 change if the data set has an odd number of data points? 15. Earlier, you found q 1 and q 3 respectively. Now represent the smallest score in the data by q 0 and the largest score in the data by q 4. You can also represent the median of the data set by q 2. Use this information to help you plot a box-and-whiskers plot. In general, what does the length of the whiskers and box tell you about a data set? Intepret the length of the whiskers and box for this data set. 16. To find a score s deviation from the mean, you simply subtraction the mean from the score. The deviation may be positive or it may be negative. Use this information to fill in the table on the following page. 17. Square each of the terms in the far right column of your table. Add all of these squares and divide by the size of the data set (30). This is called the population variance, σ 2. (So the population variance is just the mean of the squares of the deviation for each data point!) Take the square root of the population variance. This is called the population standard deviation, σ. We often prefer to look at standard deviation rather than variance; the units for the standard deviation are the units of the data, whereas the units for the variance are the units 2 from the original data! 3

4 Data Point Formula for Deviation Deviation from Mean Deviation Variance: σ 2 = Standard Deviation: σ = 18. Sketch a bell curve representing the data based on the mean and the standard deviation. 19. According to your curve, where do 68% of the scores lie? Does this match the data? 20. Based on your histogram at the beginning of the packet, do you expect conclusions drawn from the bell curve to be fairly accurate or fairly inaccurate? 21. Summarize your findings on the following page. 4

5 Summary of Findings