1. MDFP Mathematics and Statistics 1

2. Univariate Data – Today’s Class 1.STATISTICS 2.Univariate (One Variable) Data 1.Definition 2.Mean, Median, Mode, Range 3.Standard Deviation, Variance 4.Quartiles, Interquartile Range 5.Box-and-Whisker plot 3.EXAMPLES 4.Practice – E1 - 2 from Workbook 5.Conclusion Univariate Data2

3. Univariate and Bivariate Data – STATISTICS Statistics is the science of collecting, organizing and interpreting numerical and nonnumerical facts, which we call data. The collection and study of data are important in the work of many professions, so that training in the science of statistics is valuable preparation for variety of careers, for example economists and financial advisors, businessmen, engineers, farmers. Knowledge of probability and statistical methods also are useful for informatics specialists of various fields such as data mining, knowledge discovery and so on. Whatever else it may be, statistics is, first and foremost, a collection of tools used for converting raw data into information to help decision makers in their works. Univariate Data3

4. Univariate Data - Definition Univariate  In statistics, in univariate data, each data point contains only one dependent variable. The more general case is multivariate.statisticsmultivariate  This statistical method is used to evaluate single variable rather than more variables. Example: X = { 1,2,3,5,7,4,8,9,13,2,43,6,14} Univariate Data4

5. Univariate Data - Mean, Median, Mode & Range Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are the three most common. The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list. The "range" is just the difference between the largest and smallest values. Univariate Data5

6. Univariate Data - Mean, Median, Mode & Range Mean (average) Total sum divided by quantity of integers (elements, terms) Example : (34 + 26 + 45 + 31) / 4 = 34 Median Middle value that separates the greater and lesser halves of a data set Example : Odd number 13, 25, 50, 75, 80 is50 Even number 2, 3, 10, 12, 13, 20is (10+12)/2 = 11 Mode Most frequent number in a data set Example : 1, 3, 4, 4, 4, 7, 7, 12, 17is4 Range For the above example the range is 17 – 1 = 16 Univariate Data6

7. Univariate Data - Mean, Median, Mode & Range Example Find the mean, median, mode, and range for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The mean is the usual average, so: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15 Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers! Univariate Data7

8. Univariate Data - Mean, Median, Mode & Range Example The median is the middle value, so I'll have to rewrite the list in order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number: 13, 13, 13, 13, 14, 14, 16, 18, 21 So the median is 14. Univariate Data8

9. Univariate Data - Mean, Median, Mode & Range Example The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8. mean: 15 median: 14 mode: 13 range: 8 Note: The formula for the place to find the median is "( [the number of data points] + 1) ÷ 2", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer. Either way will work. Univariate Data9

10. Univariate Data - Standard Deviation, Variance Standard Deviation (sigma) - It shows how much variation there is from the "average" (mean). or Variance - of a random variable or distribution is the expected, or mean, value of the square of the deviation of that variable from its expected value or mean. Note: Always use your calculator to find the standard deviation! Univariate Data10

11. Univariate Data - Standard Deviation, Variance Example Now for the same example let us use the calculator to find the standard deviation and the variance: 13, 18, 13, 14, 13, 16, 14, 21, 13 σ n = 2.67 - standard deviation σ n 2 = 7.1289- variance Univariate Data11

12. Univariate Data - Quartiles Quartiles The median divides the data into two equal sets (halves). The lower quartile is the value of the median of the first half, where 25% of the values are smaller than Q 1 and 75% are larger. This first quartile takes the notation Q 1. The upper quartile is the value of the median of the second half, where 75% of the values are smaller than Q 3 and 25% are larger. This third quartile takes the notation Q 3. It should be noted that the median takes the notation Q 2, the second quartile. Univariate Data12

13. Univariate Data - Quartiles - Example Example 1 – Upper and lower quartiles Data 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36 Ordered data Q1 Q2 Q3 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49 Median 41 Upper quartile 43 Lower quartile 15 Univariate Data13

14. Univariate Data - Quartiles Interquartile range (IQR) Interquartile range = difference between upper quartile (Q 3 ) and lower quartile (Q 1 ) The interquartile range is another range used as a measure of the spread. The difference between upper and lower quartiles (Q 3 –Q 1 ), which is called the interquartile range, also indicates the dispersion of a data set. Univariate Data14

15. Univariate Data - Quartiles - Example Example 1 – Upper and lower quartiles Data 6, 47, 49, 15, 43, 41, 7, 39, 43, 41, 36 Ordered data 6, 7, 15, 36, 39, 41, 41, 43, 43, 47, 49 Median41 Upper quartile43 Lower quartile15 Interquartile range = Q 3 –Q 1 = 43 – 15 = 28 Univariate Data15

16. Univariate Data – Box-and-Whisker plot Example 2 Draw a box-and-whisker plot for the following data set: 4.3, 5.1, 3.9, 4.5, 4.4, 4.9, 5.0, 4.7, 4.1, 4.6, 4.4, 4.3, 4.8, 4.4, 4.2, 4.5, 4.4 Solution My first step is to order the set. This gives me: 3.9, 4.1, 4.2, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1 The first number I need is the median of the entire set. There are seventeen values in the list, so I need the 9th value: 3.9, 4.1, 4.2, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1 The median is Q 2 = 4.4. Univariate Data16

17. Univariate Data - Box-and-Whisker plot Example 2 The next two numbers I need are the medians of the two halves. Since I used the "4.4" in the middle of the list, I can't re-use it, so my two remaining data sets are: 3.9, 4.1, 4.2, 4.3, 4.3, 4.4, 4.4, 4.4 and 4.5, 4.5, 4.6, 4.7, 4.8, 4.9, 5.0, 5.1 The first half has eight values, so the median is the average of the middle two: Q 1 = (4.3 + 4.3)/2 = 4.3 The median of the second half is: Q 3 = (4.7 + 4.8)/2 = 4.75 Univariate Data17

18. Univariate Data - Box-and-Whisker plot Example 2 Since my list values have one decimal place and range from 3.9 to 5.1, I won't use a scale of, say, zero to ten, marked off by ones. Instead, I'll draw a number line from 3.5 to 5.5, and mark off by tenths. Now I'll mark off the minimum and maximum values, and Q 1, Q 2, and Q 3 : Univariate Data18

19. Univariate Data - Box-and-Whisker plot Example 2 The "box" part of the plot goes from Q 1 to Q 3 : And then the "whiskers" are drawn to the endpoints: Univariate Data19

20. Univariate Data - Practice Practice on Exercise 1 from Workbook Univariate Data20

21. Univariate Data - Conclusion What did you learn today? Why do you need to learn Univariate Data? Assignment: Any work not completed during class must be completed for homework Univariate Data21