Graphical Displays of Information 3.1 – Tools for Analyzing Data Learning Goal: Identify the shape of a histogram MSIP / Home Learning: p. 146 #1, 2, 4, 9, 11 (data in Excel file on wiki),13

Histograms Show how data is spread out Best choice for:  continuous data  discrete data with a large spread Data is divided into 5-6 intervals Bin width = width of each interval (same) Different bin widths can produce different shaped distributions

Histogram Example 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

Histogram Applet – Old Faithful 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  Incorrect: 0-10, 10-20, 20-30, 30-40, etc.  Correct: Discrete: 0 - 9, , , , etc. Concinuous: , , , , etc.

Shapes of Distributions Symmetric  Mound Shaped  U-Shaped  Uniform Unsymmetrical  Left-Skewed  Right-Skewed

Mound-shaped distribution Middle interval(s) have the greatest frequency / tallest bars Bars get shorter as you move away E.g. roll 2 dice, height, memory

U-shaped distribution  Lowest frequency in the centre, higher 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 Mound-shaped Distributions are symmetric

Skewed distribution (left or right) Highest frequencies at one end Left-skewed has higher bars on the right and drops off to the left  E.g. the years on a handful of quarters (left)  E.g. the years of cars on a classic car lot (right)

MSIP / Home Learning Define in your notes:  Frequency distribution (p )  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? modal interval? Left-skewed Modal interval: 72 – 80 Median: (70 actual) Mean: 66

Minds On! Mr. Lieff recorded the following 20 quiz marks: Find the average mark 2 different ways.

Measures of Central Tendency (Mean, Median, Mode) Chapter 3.2 – Tools for Analyzing Data Learning Goal: Calculate the mean, median and mode for weighted / grouped data Due now: p. 146 #1, 2, 4, 9, 11 (data in Excel file on wiki),13 MSIP / Home Learning: p. 159 #4, 5, 6, 8, 10-13

Sigma Notation the sigma notation is used to compactly express a mathematical series ex: … + 15 this can be expressed: 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: write in expanded form: 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) = = 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

The Mean (Average) Found by dividing the sum of all the data points by the number of data points Affected greatly by outliers Deviation (3.5)  the distance of a data point from the mean  calculated by subtracting the mean from the value  i.e.

The Weighted Mean where x i represent the data points, w i 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

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  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, Mound-shaped (could), Uniform (could) Modes are appropriate for discrete data or non-numerical data  Shoe size, Number of siblings  Eye colour, Favourite subject

Distributions and Central Tendancy the relationship between the three measures changes depending on the spread of the data symmetric (mound shaped)  mean = median = mode right skewed  mean > median > mode left skewed  mean < median < mode

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 mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7 median = 3 (27 data points, so #14 falls in bin 3) mode = 3 b) which way is it skewed? Left Survey responses 1234 Frequency 28143

Example 4 Find the mean, median and mode mean = [(145.5 × 3) + (155.5 × 7) + (165.5 × 4)] ÷ 14 = median = or mode = or MSIP / Home Learning: p. 159 #4, 5, 6, 8, Height No. of Students 374

MSIP / Home Learning p. 159 #4, 5, 6, 8, 10-13

References Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from