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 contain continuous data grouped in class intervals, which will display how data is spread over a range the width of each bar is known as the bin width different bin widths produce different shaped distributions bin widths should be equal and there should be at least five (5) 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 however, 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.isixsigma.com/offsite.asp?A=Fr&Url =http://www.stat.sc.edu/~west/javahtml/Histo gram.html Bin Width Calculation the bin width is calculated by dividing the range = max – min by the number of intervals you desire (5-6) the bins should not overlap Discrete wrong: 0-10, 10-20, 20-30, 30-40 correct: 0-10, 11-20, 21-30, 31-40 Continuous correct: 0-9.99, 10-19.99, 20-29.99, 30-39.99 Mound-shaped distribution The middle interval(s) have the greatest frequency (i.e. the tallest bar) The bars get smaller as you move out to the edges. U-shaped distribution Lowest frequency in the centre, highest towards the outside E.g. height of a combined grade 1 and 6 class 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 Normal Distributions are symmetric Skewed distribution (left and right) Highest frequencies at one end Left-skewed drops off to the left E.g. the years on a handful of quarters Exercises Define in your notes: Frequency distribution (p. 146) Cumulative frequency (p. 146) Relative frequency (p. 146) Try page 146 #1,2,3, 11 (use Excel or Fathom),13 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 1 2 3 4... 14 15 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 Examples: 7 (2n 1) n4 write in expanded form: = [2(4) + 1] + [2(5) + 1] + [2(6) + 1] + [2(7) + 1] = 9 + 11 + 13 + 15 =48 note that any letter can be used for the index of summation, though k, a, n, i, j & x are often used Example: write the following in sigma notation 3 3 3 3 2 4 8 3 3 3 3 0 1 2 3 2 2 2 2 3 2 n 0 3 n n The Mean x xi i 1 n found by dividing the sum of all the data points by the number of elements of data Deviation the distance of a data point from the mean calculated by subtracting the mean from the value 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 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) 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 Median the midpoint of the data calculated by placing all the values in order if there are an even number of values, the median is the mean of the middle two numbers 1 4 6 8 9 12 median = 7 if there is an odd number of values, the median is the middle number 1 4 6 8 9 median = 6 Mode Simply chosen by finding 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 Multiple modes? Uniform Modes are appropriate for discrete data or non-numerical data shoe sizes shoe colors 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 have the greatest effect on the mean 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, use the mode Example 1 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 mode = 3 which way is it skewed? Left Example 2 Find the mean, median and mode Height No. of Students 141-150 151-160 161-170 3 7 4 mean = [(145x3) + (155x7) + (165x4)] / 14 = 155.7 median = 155 mode = 151-160 which way is it skewed? Mound-shaped Exercises try page 159 #4, 5, 6, 8 Remembrance Day by the Numbers http://www42.statcan.ca/smr08/smr08_064_e .htm References Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page