1. 381 Descriptive Statistics-III (Measures of Central Tendency) QSCI 381 – Lecture 5 (Larson and Farber, Sects 2.3 and 2.5)

2. 381 Introduction is a value that represents the typical, or central, entry in a data set. There are three commonly used measures of central tendency: The Mean The Median The Mode.

3. 381 The Mean-I The sample mean: The population mean:

4. 381 The Mean-II Consider the data set consisting of a sample of the diameters of 6 trees in a stand: 29cm, 31cm, 43cm, 31cm, 12cm, 33cm Calculate the mean:

5. 381 The Mean-III Why we like the mean Unique. Based on every data point in the data set. Well suited to statistical treatment. Why we dislike the mean Can be sensitive to “outlying” observations.

6. 381 The Median Sort the data and average the central values. Six values: Five values: 32 31

7. 381 The Mode Find the frequency of each data entry and identify the data entry with the greatest frequency. Unlike the median and mean, the mode is not always uniquely defined. If a data set has two modes, it is referred to as being bimodal.

8. 381 Which Measure is Best? There is no clear answer to this question. The mean can be influenced by outliers while the mode may not be particularly “typical”. Statistical inference based on the median and the mode is somewhat difficult. Mode Median MeanOutlier?

9. 381 Computing the Mean of a Group of Data Points Suppose the data are in the form of frequencies, i.e., for each i, we have x i and f i, where f i is number of data entries for which x equals x i, then: In Excel use: “sumproduct(a1:a10,b1:b10)/sum(b1:b10)” where the x i ’s are stored in column A and the f i ’s are stored in column B.

10. 381 Shapes of Distributions-I A frequency distribution is when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are mirror images. Mean, Median, Mode

11. 381 Shapes of Distributions-II A frequency distribution is (or rectangular) when the number of entries in each class is equal (a uniform distribution is symmetric). Mean, Median, Mode

12. 381 Shapes of Distributions-III A frequency distribution is (or positively skewed) if its tail extends to the right (mode < median < mean). Mean Tail Mode Median

13. 381 Shapes of Distributions-IV A frequency distribution is (or negatively skewed) if its tail extends to the left (mode > median > mean).

14. 381 Fractiles The is the difference between the maximum and minimum data entries. The : Q 1, Q 2, and Q 3, divide a (ordered) data set into four equal parts. The : P 1, P 2, ….P 99 divide a (ordered) data set into 100 equal parts. Collectively, Quartiles, Percentiles (and Deciles) are referred to as Fractiles.

15. 381 More on Quartiles The quartiles divide a data set at the 25 th percentile, the 50 th percentile, and the 75 th percentile. The 50 th percentile is the median. The difference between the 75 th and 25 th percentiles is referred to as the.

16. 381 More on Percentiles 80% 15.2m Interpretation: 80% of the bowheads caught are smaller than 15.2m

17. 381 Box and Whisker Plots-I The information on the range and the quartiles can be represented using a box and whisker plot.

18. 381 Box and Whisker Plots-II Find the five number summary of the data (range, Q 1,Q 2,Q 3 ). Construct a horizontal line that spans the data. Plot the five numbers above the horizontal scale. Draw a box above the horizontal scale from Q 1 to Q 3 and draw a vertical line in the box at Q 2. Minimum MaximumQ1Q1 Q 2 Median Q3Q3 whisker 15105 Length (m)

19. 381 Review of Symbols in this Lecture