Recap

1. Recap • Review the central tendencies – Mean, Median, Mode • Understand the different types of statistical representations

2. Central Tendencies - 3 M’s • Mean - the arithmetic mean, • is commonly called the average. • Median - the middle data/score • the middle of a distribution: half the scores are above the median and half are below the median. • Mode • is the observation that occurs most frequently in the sample.

3. Example (Mean, Median & Mode) • Consider a XXX company, • The CEO makes $100,000 per year, • Two managers make $50,000 per year, • Fifteen factory workers make $15,000 each • Two trainees make $9,000 per year. • Mean = $22,150 • But only three out of 20 persons having salary greater than $22K! • Median = $15,000 • Mode = $15,000 • What about Measure of Spread?

4. Another Example A professor of statistics wants to report the results of a midterm exam, taken by 100 students. The mean provides information about the over-all performance level of the class. It can serve as a tool for making comparisons with other classes and/or other exams. The mode must be used when data is qualitative. If marks are classified by letter grade, the frequency of each grade can be calculated.Then, the mode becomes a logical measure to compute.

5. Example (cont’) The Median indicates that half of the class received a grade below 81%, and half of the class received a grade above 81%.

6. Interesting Facts about 3 M’s • Looking at their relationship • If a distribution is symmetrical, the mean, median and mode coincide

7. A negatively skewed distribution (“skewed to the left”) A positively skewed distribution (“skewed to the right”) Mode Mean Mode Mean Median Median Interesting Facts about 3 M’s • If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ.

8. Interesting Facts about 3 M’s • Mean is called economic indicator. It can provide sample total. • Medianis called social indicator. It can better identify a typical value. • Mean of a subgroup can be integrated to get the total mean, but not median. • Trimmed mean is an alternative mean that is resistant to outliers. 

9. Example 1 • Given a set of 11 numbers: 3, 5, 5, 5, 6, 7, 7, 8, 8, 9, and 14, find the mean, median and mode of this set of data. • Mean: 7 • Median: 7 • Mode: 5

10. How to describe the data? • Skewed to the Right / Left ? • Any possible outliers?

11. Graphical Presentation of Qualitative Data • Pie Chart • Bar Chart • Line Chart

12. 170cm 150cm 160cm 180cm 140cm Graphical Presentation of Quantitative Data • Dot Diagram • useful in displaying a small body of data, say up to about 20 observations. • example : height of students in a class.

13. -200 -100 100 0 200 Another Example (Changi International Airport) • A study at the Changi airport regarding the flight delay times for 19 flights are shown as below: -8 -6 180 10 5 -15 20 -49 28 12 32 -23 29 -19 2 35 13 11 9

14. Graphical Presentation of Quantitative Data • Stem-and-leaf Diagram • a good way to obtain informative visual display of data. • we divide each number into two part: a stem, consisting of one or more leading digit and leaf, consisting of remaining digits. • Example (height) StemLeaf Frequency 148 1 152 5 6 2 4 161 2 3 7 8 6 9 7 171 2 7 5 6 5 182 1

15. Variance and standard Deviation • Measure of Spread • Box-Plot • Understand and Use Variance and Standard Deviation

16. Measure of Spread • Range • the difference between the largest and the smallest values • Highest Value - Lowest Value • Interquartile Range • the difference between the 75th percentile (often called Q3) and the 25th percentile (Q1). • Q3-Q1 • Standard Deviation, • the square root of the variance.

17. So, what is variance, ? • a measure of how spread out a distribution is • It is computed as the average squared deviation of each number from its mean. • For example, for the numbers 1, 2, and 3, the mean is 2 and the variance is:

18. 3 4 5 6 7 8 9 10 11 12 13 14 Box Plot

19. Box Plot • is sometimes also called the Five Number Summary.

20. Low High Using Box Plots • Bigger the spread, bigger the magnitude of the range, interquartile range, and standard deviation

21. Example 2 • A group of boys (HCI-2Mers) score an average of 75% in a test while the girls (NYG-2Modest) obtained an average of 70%. On the other hand, the standard deviation of the scores for the boys is 10% while that of the girls is only 5%. In your opinion, which group has done better? Why?

22. Some Important Information

23. Computing Variance • Step 1: Obtain sum of squares • Step 2: Obtain mean of data • Step 3: Use

24. Example 3 • Find the mean, variance and the standard deviation for the set of data {24, 27, 27, 30, 31, 31, 31, 35, 36, 38} • Mean: 31 • Variance: 18.2 • Standard deviation: 4.266

25. Example 4 • 200 people were surveyed to indicate the number of days, X days, they had dinner at home with their families last week. The following shows the frequency table for the data collected.

26. Example 4 • Find the mean, variance and standard deviation of X. • Mean: 3.075 • Variance: 2.76 • Standard deviation: 1.66

27. Additonal Information

28. Example 3 • Find the mean, variance and the standard deviation for the set of data {24, 27, 27, 30, 31, 31, 31, 35, 36, 38} • Mean: 31 • Variance: 18.2 • Standard deviation: 4.266

29. Computing Variance • Step 1: Obtain sum of squares • Step 2: Obtain mean of data • Step 3: Use

30. Example 4 • Find the mean, variance and standard deviation of X. • Mean: 3.075 • Variance: 2.76 • Standard deviation: 1.66