chapter no. 2. describing central tendency and variability .ppt
- 1. Chapter No. 2 Tools for describing central tendency and variability in data.
- 2. Learning Objectives • Distinguish between measures of central tendency, measures of variability, and measures of shape • Understand the meanings of mean, median, mode, quartile, and range • Compute mean, median, mode, quartile, range, variance, standard deviation, and mean absolute deviation
- 3. Learning Objectives -- Continued • Differentiate between sample and population variance and standard deviation • Understand the meaning of standard deviation as it is applied by using the empirical rule • Understand box and whisker plots, skewness, and kurtosis
- 4. Measures of Central Tendency • Measures of central tendency yield information about “particular places or locations in a group of numbers.” • Common Measures of Location –Mode –Median –Mean –Quartiles
- 5. Mode • The most frequently occurring value in a data set • Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) • Bimodal -- Data sets that have two modes • Multimodal -- Data sets that contain more than two modes
- 6. • The mode is 44. • There are more 44s than any other value. 35 37 37 39 40 40 41 41 43 43 43 43 44 44 44 44 44 45 45 46 46 46 46 48 Mode -- Example
- 7. Median • Middle value in an ordered array of numbers. • Applicable for ordinal, interval, and ratio data • Not applicable for nominal data • Unaffected by extremely large and extremely small values.
- 9. Median: Computational Procedure • First Procedure – Arrange observations in an ordered array. – If number of terms is odd, the median is the middle term of the ordered array. – If number of terms is even, the median is the average of the middle two terms. • Second Procedure – The median’s position in an ordered array is given by (n+1)/2.
- 10. Median: Example with an Odd Number of Terms Ordered Array includes: 3, 4, 5, 7, 8, 9, 11, 14, 15, 16 ,16, 17, 19, 19, 20, 21, 22 • There are 17 terms in the ordered array. • Position of median = (n+1)/2 = (17+1)/2 = 9 • The median is the 9th term, 15. • If the 22 is replaced by 100, the median remains at 15. • If the 3 is replaced by -103, the median remains at 15.
- 11. Mean • Is the average of a group of numbers • Applicable for interval and ratio data, not applicable for nominal or ordinal data • Affected by each value in the data set, including extreme values • Computed by summing all values in the data set and dividing the sum by the number of values in the data set
- 12. Population Mean X N N X X X XN 1 2 3 24 13 19 26 11 5 93 5 18 6 ... .
- 13. Sample Mean X X n n X X X Xn 1 2 3 57 86 42 38 90 66 6 379 6 63 167 ... .
- 15. • Say you have a data set of : 4, 5, 5, 6, 6, 6, 7, 7, 80 4, 5, 5, 6, 6, 6, 7, 7, 80 • The mean is 14 14. (When you take the sum and divide by how many pieces of data there are, you get 126/9 which equals 14) You now have a data set of : 4, 5, 5, 6, 6, 6, 7, 7 4, 5, 5, 6, 6, 6, 7, 7 •The mean is 5.75 5.75, this is a big change from 14. (When you take the sum and divide by how many pieces of data there are, you get 46/8 which equals 5.75) extreme values have large effects on the mean.
- 16. Quartiles Measures of central tendency that divide a group of data into four subgroups • Q1: 25% of the data set is below the first quartile • Q2: 50% of the data set is below the second quartile • Q3: 75% of the data set is below the third quartile
- 17. Quartiles 25% 25% 25% 25% Q 3 Q 2 Q 1
- 18. The median divides the data into a lower half and an upper half. The lower quartile is the middle value of the lower half. The upper quartile is the middle value of the upper half.
- 19. Box & Whisker Box & Whisker Plots Plots
- 20. What is a Box & Whisker Plot What is a Box & Whisker Plot: : Box & Whisker Plots Box & Whisker Plots Lower Extreme (MIN) Lower Quartile (Q1) Median (Q2) Upper Quartile (Q3) Upper Extreme (MAX)
- 21. Vocabulary Vocabulary: : Box & Whisker Plot: Box & Whisker Plot: a diagram that summarizes data using the a diagram that summarizes data using the median, the upper and lower quartiles, and the extreme values. median, the upper and lower quartiles, and the extreme values. Upper Extreme: Upper Extreme: the greatest value in a set of data. the greatest value in a set of data. Lower Extreme: Lower Extreme: the smallest value in a set of data. the smallest value in a set of data. Upper Quartile: Upper Quartile: the median of the upper half of a set of data; the median of the upper half of a set of data; UQ. UQ. Lower Quartile: Lower Quartile: the median of the lower half of a set of data; the median of the lower half of a set of data; LQ. LQ. Interquartile Range: Interquartile Range: the range of the middle half of a set of the range of the middle half of a set of data; UQ – LQ. data; UQ – LQ. Box & Whisker Plots Box & Whisker Plots
- 22. One More Example One More Example: : Box & Whisker Plots Box & Whisker Plots
- 23. Why Box & Whisker Plots Why Box & Whisker Plots: : - Box and whisker plots use the median, upper - Box and whisker plots use the median, upper and lower quartiles, and the extreme (least and and lower quartiles, and the extreme (least and greatest) values to summarize data. greatest) values to summarize data. - It allows you to see important characteristics of - It allows you to see important characteristics of the data at a glance. the data at a glance. - Even though they can be difficult to create, they - Even though they can be difficult to create, they can provide a great amount of important can provide a great amount of important information. information. Box & Whisker Plots Box & Whisker Plots
- 24. Box & Whisker Plots Box & Whisker Plots Example 1 Example 1: : Nutrition Facts Nutrition Facts The grams of fat per serving of items from the meat, The grams of fat per serving of items from the meat, poultry, and fish food group are shown in the poultry, and fish food group are shown in the table. Make a box-and-whisker plot of the data. table. Make a box-and-whisker plot of the data. 7 Tuna 18 Ground Beef 9 Trout 10 Fried Shrimp 9 Sardines 3 Fish Sticks 5 Salmon 3 Crabmeat 5 Roast Beef 16 Bologna 19 Pork Chop 15 Beefsteak 14 Ham 9 Bacon Fat (gm) Item Fat (gm) Item
- 25. Box & Whisker Plots Box & Whisker Plots Example 1 Example 1: : Nutrition Facts Nutrition Facts 3, 3, 5, 5, 7, 9, 9, 9, 10, 14, 15, 16, 18, 19 3, 3, 5, 5, 7, 9, 9, 9, 10, 14, 15, 16, 18, 19 Step 2: Step 2: Find the Median, L/U Quartiles, and L/U Extremes. Find the Median, L/U Quartiles, and L/U Extremes. Step 1: Step 1: Order the data from least to greatest. Order the data from least to greatest. Median: 9 Median: 9 Lower Quartile Lower Quartile is median of is median of lower half = 5 lower half = 5 Upper Quartile is Upper Quartile is median of upper median of upper half = 15 half = 15 Lower Lower Extreme: 3 Extreme: 3 Upper Upper Extreme: 19 Extreme: 19 Fat Fat Fat Fat 9 9 14 14 15 15 19 19 16 16 5 5 3 3 5 5 3 3 9 9 10 10 9 9 18 18 7 7
- 26. Box & Whisker Plots Box & Whisker Plots Example 1 Example 1: : Nutrition Facts Nutrition Facts Step 4: Step 4: Find the median and the quartiles. Find the median and the quartiles. Step 3: Step 3: Draw a number line. The scale should include Draw a number line. The scale should include the median, the quartiles, and the lower and the median, the quartiles, and the lower and upper extremes. Graph the values as points upper extremes. Graph the values as points above the above the. .
- 27. Box & Whisker Plots Box & Whisker Plots Example 1 Example 1: : Nutrition Facts Nutrition Facts The graph shows that half of the foods have between 3 and 9 grams of fat. The largest range of the four quartiles is from 9 to 15. One-fourth of the foods were within this range.
- 28. Independent Practice Problems Independent Practice Problems: : Box & Whisker Plots Box & Whisker Plots The table below shows the commute time from home to school for fifteen middle school students. Make a box-and-whisker plot of the data. Then use it to describe how the data are spread. Student Commute Time 25 14 7 18 10 46 21 5 11 18 23 17 9 13 12 The graph shows that half of the students travel between 10 and 21 minutes. The largest range of the four quartiles is from 21 to 46.
- 29. Why Calculate Outliers Why Calculate Outliers: : Box & Whisker Plots Box & Whisker Plots - First, remember that statistics assume all our data values are clustered around some central value - For a box and whisker plot, the IQR tells how spread out the "middle" values are - But it can also be used to tell when some of the other values are "too far" from the central value - Any value that is "too far away" is called an "outlier," because it "lies outside" the range in which we expect it to be
- 30. Box & Whisker Plots Box & Whisker Plots Example 2 Example 2: : 396, 410, 487, 627, 645, 648, 710, 731, 801, 894 396, 410, 487, 627, 645, 648, 710, 731, 801, 894 Looking back at this data set we can Looking back at this data set we can determine if any outliers exist determine if any outliers exist Median: 646.5 Median: 646.5 Lower Quartile is Lower Quartile is Median of lower half = 487 Median of lower half = 487 Upper Quartile is Upper Quartile is Median of upper half = 731 Median of upper half = 731 NO. of NO. of patient patient NO. of NO. of patient patient 894 894 645 645 801 801 717 717 731 731 410 410 627 627 648 648 487 487 396 396 Lower Lower Extreme: 396 Extreme: 396 Upper Upper Extreme: 894 Extreme: 894
- 31. Box & Whisker Plots Box & Whisker Plots Example 2 Example 2: : Looking back at this data set we can Looking back at this data set we can determine if any outliers exist. determine if any outliers exist. NO. of NO. of patient patient NO. of NO. of patient patient 894 894 645 645 801 801 717 717 731 731 410 410 627 627 648 648 487 487 396 396 LE = 396 LE = 396 LQ (Q LQ (Q1 1) = 487 ) = 487 M = 646.5 M = 646.5 UQ (Q UQ (Q3 3) = 731 ) = 731 UE = 894 UE = 894 IQR IQR = Q3 – Q1 = Q3 – Q1 IQR IQR = 731 – 487 = 731 – 487 IQR IQR = 244 = 244
- 32. Box & Whisker Plots Box & Whisker Plots Example 3 Example 3: : Box & Whisker Plots (Additional) Box & Whisker Plots (Additional) Use a stem and leaf plot to graph the following test Use a stem and leaf plot to graph the following test scores from a recent scores from a recent statistic statistic test : test : 76, 76, 76, 77, 80, 80, 80, 81, 81, 82, 84, 85, 88, 89, 89, 76, 76, 76, 77, 80, 80, 80, 81, 81, 82, 84, 85, 88, 89, 89, 89, 89, and 92 89, 89, and 92
- 33. HOMEWORK HOMEWORK Box & Whisker Plots Box & Whisker Plots
- 34. Box & Whisker Plots Box & Whisker Plots Example 5 Example 5: : Applying Box & Whisker Plots Applying Box & Whisker Plots Now lets use the Percents Test Data to create a Box and Now lets use the Percents Test Data to create a Box and Whisker Plot Whisker Plot of the data and then use it to describe the spread of the data and then use it to describe the spread of the data of the data. . BOYS SCORES BOYS SCORES 99, 36, 16, 23, 69, 58, 59, 21, 53, 19, 21, 82, 30, 85, 70, 81, 66, 42, 53, 52, 22, 56, 43, 57, 88, 80, 53, 86, 64, 84, 68, 79, 57, and 82 GIRLS SCORES GIRLS SCORES 73, 37, 61, 53, 37, 38, 24, 30, 75, 93, 65, 85, 60, 92, 80, 56, 80, 65, 77, 64, 95, 99, 82, 75, 94, 98, 63, 58, 69, 56, 95, 77, and 45
- 35. Stem and Leaf Plots/Diagrams A stem and leaf plot is a frequency diagram in which the raw data is displayed together with its frequency. The data is then placed in suitable groups. Example. The following data gives the marks out of 60 for a maths test. Place the data in a stem and leaf diagram. 28, 38, 42, 5, 13, 23, 14, 38, 56, 20, 32, 47, 58, 3, 18, 19, 42, 48, 29, 14, 24, 28, 50, 31, 35 Stem Leaf 0 1 2 3 4 5 3 5 3 4 4 8 9 0 3 4 8 1 5 8 8 2 2 7 8 8 9 0 6 8 2 0 – 9 10 - 19 20 – 29 30 – 39 40 – 49 50 – 59 tens units Median is the 13th data value
- 36. Stem and Leaf Plots/Diagrams The stem and leaf plot below shows the masses in kg of some people in a lift. (a) How many people were weighed? (b) What is the range of the masses? (c) Find the median mass. Stem Leaf 3 4 5 6 7 8 1 4 3 3 6 0 3 4 8 1 2 2 7 1 6 2 tens units (a) 16 people. (b) 86 – 31 = 55 kg Median is the mean of the 8th and 9th data values. (c) 56 kg
- 37. High temperatures for the last week: 72, 78, 87, 90, 88, 86, 87, 89 Stem 7 8 9 Leaf 2 8 6 7 7 8 9 0 7 2 = 72 degrees
- 38. Using Stem-and-Leaf Plots • What was the least number of cookies sold? Number of isolated strains of bacteria from skin surface samples. 40 strains
- 39. Using Stem-and-Leaf Plots • What was the largest number of cookies sold? Number of isolated strains of bacteria from skin surface samples. 94 strains
- 40. Using Stem-and-Leaf Plots • Find the median of the data Find the middle leaf… 63 strains Number of isolated strains of bacteria from skin surface samples.
- 41. Using Stem-and-Leaf Plots • Find the mode of the data Look for the number that occurs most often in a row of leaves 63 strains Boxes of Girl Scout Cookies Sold by Troop 220
- 42. Stem & Leaf vs. Box & Whisker Stem & Leaf vs. Box & Whisker Leaf (Girls Data) Stem Leaf (Boys Data) 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 4 4 0 7 7 8 0 7 7 8 5 5 3 6 6 8 3 6 6 8 0 1 3 4 5 5 9 0 1 3 4 5 5 9 3 5 5 7 7 3 5 5 7 7 0 0 2 5 0 0 2 5 2 3 4 5 5 8 9 2 3 4 5 5 8 9 9 6 9 6 3 2 1 1 3 2 1 1 6 0 6 0 3 2 3 2 9 8 7 7 6 3 3 3 2 9 8 7 7 6 3 3 3 2 9 8 6 4 9 8 6 4 9 0 9 0 8 6 5 4 2 2 1 0 8 6 5 4 2 2 1 0 9 9