Exploring Data. Kwonsang Lee. January 30, University of Pennsylvania Kwonsang Lee STAT111 January 30, / 12

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1 Exploring Data Kwonsang Lee University of Pennsylvania January 30, 2015 Kwonsang Lee STAT111 January 30, / 12

2 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: Kwonsang Lee STAT111 January 30, / 12

3 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: = 4 Kwonsang Lee STAT111 January 30, / 12

4 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: = 4 Median: Kwonsang Lee STAT111 January 30, / 12

5 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: = 4 Median: = 4 Kwonsang Lee STAT111 January 30, / 12

6 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: = 4 Median: = 4 Mode: Kwonsang Lee STAT111 January 30, / 12

7 Mean, Median and Mode There are six observations; 1, 1, 2, 6, 6, 8 Mean: = 4 Median: = 4 Mode: 1, 6 Kwonsang Lee STAT111 January 30, / 12

8 Quartiles Quartiles: the first quartile Q 1, the second quartile=median M, the third quartile Q 3 The first quartile Q 1 : divides into the lowest 25% and the highest 75% Median M: divides into the lowest 50% and the highest 50% The third quartile Q 3 : divides into the lowest 75% and the highest 25% Interquartile range (IQR): IQR= Q 3 Q 1 Kwonsang Lee STAT111 January 30, / 12

9 Boxplot Kwonsang Lee STAT111 January 30, / 12

10 Histogram Histogram is a good way to see the shape of a distribution. Kwonsang Lee STAT111 January 30, / 12

11 Histogram Choosing bin width (or the number of bins) is an important issue! Kwonsang Lee STAT111 January 30, / 12

12 Question 1 The normal distribution is unimodal - that is, it has only one mode or peak. Not all distributions are unimodal: some may have two or more peaks. Suppose you make a boxplot of a dataset that has two peaks. Could you tell just from looking at the boxplot that the data are not unimodal? Answer: Kwonsang Lee STAT111 January 30, / 12

13 Question 1 The normal distribution is unimodal - that is, it has only one mode or peak. Not all distributions are unimodal: some may have two or more peaks. Suppose you make a boxplot of a dataset that has two peaks. Could you tell just from looking at the boxplot that the data are not unimodal? Answer: No. Kwonsang Lee STAT111 January 30, / 12

14 Example: Boxplots of set1 and set2 Which one is a unimodal distribution? Both have mean 0 and sd 5.4 Kwonsang Lee STAT111 January 30, / 12

15 Example: Histograms of set1 and set2 Kwonsang Lee STAT111 January 30, / 12

16 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 Kwonsang Lee STAT111 January 30, / 12

17 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 Kwonsang Lee STAT111 January 30, / 12

18 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 Kwonsang Lee STAT111 January 30, / 12

19 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 Kwonsang Lee STAT111 January 30, / 12

20 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 Kwonsang Lee STAT111 January 30, / 12

21 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 2 Kwonsang Lee STAT111 January 30, / 12

22 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 2 d. 54, 49, 46, 49, 51 Kwonsang Lee STAT111 January 30, / 12

23 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 2 d. 54, 49, 46, 49, 51 2(3) Kwonsang Lee STAT111 January 30, / 12

24 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 2 d. 54, 49, 46, 49, 51 2(3) e. 60, 36, 31, 50, 48 Kwonsang Lee STAT111 January 30, / 12

25 Question 2 Each of the following lists has an average of 50. For each one, state whether the standard deviation is approximately 1, 2 or 10. This does not require the use of a calculator. a. 49, 51, 49, 51, 49 1 b. 48, 52, 48, 52, 48 2 c. 48, 51, 49, 52, 47 2 d. 54, 49, 46, 49, 51 2(3) e. 60, 36, 31, 50, Kwonsang Lee STAT111 January 30, / 12

26 Question 3 The histograms on the backside of this paper summarize data that was taken from a research paper reporting on an experimental study of drugs to relieve itching. Reported here are the results from two of the drugs, Papaverine(Papaver) and Pentobarbital (Pentob), and from a third control (Placebo) group. Each group contained 10 males aged which were given their drug intravenously, and then itching was induced on their forearms with cowage, an effective itch stimulus. The subjects recorded the duration of the itching (in seconds). Kwonsang Lee STAT111 January 30, / 12

27 Question 3 a. One definition of the center of the distribution is the value that splits the histogram in half. Approximately what is the center of the distribution for the Pentobarbital group? answer the same question for the Papaverine and Placebo groups. b. Indentify any outliers (any observations that are suspiciously large or small) c. What is the shape of each of the three histograms (symmetric, skew to the right, skew to the left) d. discuss what are the conclusions you can draw from this data as to the effectiveness of these treatments. e. Concerning the placebo group an alternate suggestion had been to save money by not giving these subjects an intravenous sugar solution. Why would this not have been a good idea? Kwonsang Lee STAT111 January 30, / 12