Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Graphical Displays of Information Chapter 3.1 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U Histograms Show: Continuous data grouped in class intervals How data is spread over a range Bin width = width of each bar Different bin widths produce different shaped distributions Bin widths should be equal Usually 5-6 bins Histogram Example Histogram Data 9 8 7 Count 6 5 4 3 2 Histogram Data 1 30 40 25 60 80 SomeData 100 20 15 10 5 40 60 80 SomeData 100 120 Data Histogram 6 5 Count These histograms represent the same data One shows much less of the structure of the data Too many bins (bin width too small) is also a problem Count 4 3 2 1 30 40 50 60 70 80 SomeData 90 100 110 120 Histogram Applet – Old Faithful http://www.stat.sc.edu/~west/javahtml/Histogra m.html Bin Width Calculation Bin width = (range) ÷ (number of intervals) where range = (max) – (min) Number of intervals is usually 5-6 Bins should not overlap wrong: 0-10, 10-20, 20-30, 30-40, etc. Discrete correct: 0-10, 11-20, 21-30, 31-40, etc. correct: 0-10.5, 10.5-20.5, 20.5-30.5, etc. Continuous correct: 0-9.9, 10-19.9, 20-29.9, 30-39.9, etc. correct: 0-9.99, 10-19.99, 20-29.99, 30-39.99, etc. Mound-shaped distribution The middle interval(s) have the greatest frequency (i.e. the tallest bars) The bars get shorter as you move out to the edges. E.g. roll 2 dice 75 times U-shaped distribution Lowest frequency in the centre, higher towards the outside E.g. height of a combined grade 1 and 6 class Student Heights 12 10 8 Frequency 6 4 2 0 Height (cm) Uniform distribution All bars are approximately the same height e.g. roll a die 50 times Symmetric distribution A distribution that is the same on either side of the centre U-Shaped, Uniform and Mound-shaped Distributions are symmetric Skewed distribution (left or right) Highest frequencies at one end Left-skewed drops off to the left E.g. the years on a handful of quarters MSIP / Homework Define in your notes: Frequency distribution (p. 142-143) Cumulative frequency (p. 148) Relative frequency (p. 148) Complete p. 146 #1, 2, 4 , 9, 11 (data in Excel file on wiki),13 Warm up - Class marks What shape is this distribution? Which of the following can you tell from the graph: mean? median? mode? Left-skewed Mean < median < mode Modal interval: 76 (Median: 70) (Mean: 66) Measures of Central Tendency Chapter 3.2 – Tools for Analyzing Data Mathematics of Data Management (Nelson) MDM 4U Sigma Notation the sigma notation is used to compactly express a mathematical series ex: 1 + 2 + 3 + 4 + … + 15 this can be expressed: 15 k k 1 the variable k is called the index of summation. the number 1 is the lower limit and the number 15 is the upper limit we would say: “the sum of k for k = 1 to k = 15” Example 1: 7 write in expanded form: (2n 1) n4 This is the sum of the term 2n+1 as n takes on the values from 4 to 7. = (2×4 + 1) + (2×5 + 1) + (2×6 + 1) + (2×7 + 1) = 9 + 11 + 13 + 15 = 48 NOTE: any letter can be used for the index of summation, though a, n, i, j, k & x are the most common Example 2: write the following in sigma notation 3 3 3 3 2 4 8 3 3 n n 0 2 n The Mean x x i 1 i n Found by dividing the sum of all the data points by the number of elements of data Affected greatly by outliers Deviation the distance of a data point from the mean calculated by subtracting the mean from the value i.e. x x The Weighted Mean n x xw i 1 n i w i 1 i i where xi represent the data points, wi represents the weight or the frequency “The sum of the products of each item and its weight divided by the sum of the weights” see examples on page 153 and 154 example: 7 students have a mark of 70 and 10 students have a mark of 80 mean = (70×7 + 80×10) ÷ (7+10) = 75.9 Means with grouped data for data that is already grouped into class intervals (assuming you do not have the original data), you must use the midpoint of each class to estimate the weighted mean see the example on page 154-5 and today’s Example 4 Median the midpoint of the data calculated by placing all the values in order if there is an odd number of values, the median is the middle number median = 6 if there are an even number of values, the median is the mean of the middle two numbers 1 4 6 8 9 1 4 6 8 9 12 median = 7 not affected greatly by outliers Mode The number that occurs most often There may be no mode, one mode, two modes (bimodal), etc. Which distributions from yesterday have one mode? Mound-shaped, Left/Right-Skewed Two modes? U-Shaped, some Symmetric Modes are appropriate for discrete data or non-numerical data Eye colour Favourite Subject Distributions and Central Tendancy the relationship between the three measures changes depending on the spread of the data Histogram Data symmetric (mound shaped) Count 3 mean = median = mode 2 1 Histogram Data 0 5 right skewed mean > median > mode 2 3 4 data 5 6 7 4 Count 1 3 2 1 Histogram Data 0 1 2 3 4 data 5 6 7 5 left skewed mean < median < mode Count 4 3 2 1 0 1 2 3 4 data 5 6 7 What Method is Most Appropriate? Outliers are data points that are quite different from the other points Outliers affect the mean the greatest The median is least affected by outliers Skewed data is best represented by the median If symmetric either median or mean If not numeric or if the frequency is the most critical measure, use the mode Example 3 a) Find the mean, median and mode Survey responses 1 2 3 4 Frequency 2 8 14 3 mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7 median = 3 (27 data points, so #14 falls in bin 3) mode = 3 b) What shape does it have? Left-skewed Example 4 Find the mean, median and mode Height No. of Students 141-150 151-160 161-170 3 7 4 mean = [(145.5×3) + (155.5×7) + (165.5×4)] ÷ 14 = 156.2 median = 151-160 or 155.5 mode = 151-160 or 155.5 MSIP / Homework: p. 159 #4, 5, 6, 8, 10-13 MSIP / Homework p. 159 #4, 5, 6, 8, 10-13 References Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page