Statistics: Analyzing Data by Using Tables and Graphs 1. 8; 1. 9; 5 CCSS: N-Q (1-3); S-ID 1

Mathematical Practice 1. Make sense of problems, and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments, and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for, and make use of, structure. 8. Look for, and express regularity in, repeated reasoning.

Statistics- Definitions A population is the collection of all the data that could be observed in a statistical study. A sample is a collection of data chosen from the population of interest. It is some smaller portion of the population. An inference is a decision, estimate, prediction, or generalization about a population based on information contained in a sample from that population.

Statistics- Examples Population All NCU students All voters enrolled during in the 2004 summer 2004 election Sample 500 NCU students 2500 voters enrolled during in the 2004 Inference The mean time to About 45% drive to NCU is of voters 24 minutes favor Amanda.

SHAPES Skewed Right: Most of the data is concentrated to the left of the graph (tail point to the right) Skewed Left Most of the data is concentrated to the right of the graph (tail points to the left) Symmetric: The majority of the data is concentrated in the center of the graph (shaped like a bell)

Center and Spread Center: the value that divides the observations so that about half have smaller values Spread: the smallest and larges values expressed in an interval

The Arithmetic Mean This is the most popular and useful measure of central location Sum of the observations Number of observations Mean = This is often called the average.

Useful Notation x: lowercase letter x - represents any measurement in a sample of data. n: lowercase letter n – number of measurements in a sample ∑: uppercase Greek letter sigma – represents sum ∑x: - add all the measurements in a sample. : – lowercase x with a bar over it – denotes the sample mean

Measures of Center 1) Sample Mean: where n is the sample size. 2) Sample Median: First, put the data in order. Then, the middle number for odd sample sizes median = the average of the two middle values for even sample sizes

The Arithmetic Mean Example 1 The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet. 7 22 11.0

The Median The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude. . Find the median of the time on the internet for the 10 adults of example 3.1 Suppose only 9 adults were sampled (exclude, say, the longest time (33)) Comment 0, 0, 5, 7, 8, 9, 12, 14, 22, 33 Even number of observations Odd number of observations 0, 0, 5, 7, 8, 9, 12, 14, 22, 33 8.5, 0, 0, 5, 7, 8 9, 12, 14, 22 8

Examples – Time to Complete an Exam A random sample of times, in minutes, to complete a statistics exam yielded the following times. Compute the mean and median for this data. 33, 29, 45, 60, 42, 19, 52, 38, 36 The mean is minutes Recall, we must rank (sort) the data before finding the median. 19, 29, 33, 36, 38, 42, 45, 52, 60 Since there are 9 (odd) data points, the 5th point is the median. The median is 38 minutes.

Examples – Miles Jogged Last Week A random sample of 12 joggers were asked to keep track of the distance they ran (in miles) over a week’s time. Compute the mean and median for this data. 5.5, 7.2, 1.6, 22.0, 8.7, 2.8, 5.3, 3.4, 12.5, 18.6, 8.3, 6.6 miles

Examples – Miles Jogged Last Week (Cont) A random sample of 12 joggers were asked to keep track of the distance they ran (in miles) over a week’s time. Compute the mean and median for this data. 5.5, 7.2, 1.6, 22.0, 8.7, 2.8, 5.3, 3.4, 12.5, 18.6, 8.3, 6.6 Recall, we must rank (sort) the data before finding the median. 1.6, 2.8, 3.4, 5.3, 5.5, 6.6, 7.2, 8.3, 8.7, 12.5, 18.6, 22.0 Since there are 12 (even) data points, the median is the average of the 6th and 7th points. The median is 6.9 miles.

Algebra 1-9 Statistics - Analyzing Data by Using Tables and Graphs A bar graph compares different categories of numerical information, or data, by showing each category as a bar whose length is related to the frequency. Bar graphs can also be used to display multiple sets of data in different categories at the same time. Graphs with multiple sets of data always have a key to denote which bars represent each set of data. Harbour

Vocabulary Bar graph: compares different categories of numerical information, of data.

Algebra 1-9 Statistics - Analyzing Data by Using Tables and Graphs Another type of graph used to display data is a circle graph. A circle graph compares parts of a set of data as a percent of the whole set. The percents in a circle graph should always have a sum of 100%. Harbour

Circle graph: compares parts of a set of data as a percent of the whole set. National Traffic Survey 3% Not sure 26% Same 8% Better 63% Worse

Algebra 1-9 Statistics - Analyzing Data by Using Tables and Graphs Another type of graph used to display data is a line graph. Line graphs are useful when showing how a set of data changes over time. They can also be helpful when making predictions. Harbour

Cable Television Systems, Line graph: numerical data displayed to show trends or changes over time. Cable Television Systems, 1995-2000 11.2 11.0 10.8 Systems (in thousands) 10.6 10.4 10.2 ‘95 ‘96 ‘97 ‘98 ‘99 ‘00 Year

Algebra 1-9 Statistics - Analyzing Data by Using Tables and Graphs Type of Graph Bar graph Circle graph Line graph When to Use To compare different categories of data To show data as parts of a whole set of data To show the change in data over time Harbour

Frequency Chart A Frequency Chart is a table that breaks data down into equal intervals and then counts the amount data in each interval. A Frequency Chart is often used to sort a list of data to make a Histogram. Make a Frequency Chart to display the data below: 90, 85, 78, 55, 64, 94, 68, 83, 84, 71, 74, 75, 99, 52, 98, 84, 73, 96, 81, 58, 97, 75, 80, 78 Interval 50-59 60-69 70-79 80-89 90-99 Frequency of Data 3 2 7 6 6

Don’t forget little things…like labels and equal intervals! Creating a Histogram Don’t forget little things…like labels and equal intervals! Interval 50-59 60-69 70-79 80-89 90-99 Frequency of Data 3 2 7 6 10 Math Test Scores 8 6 Frequency 4 2 50-59 60-69 70-79 80-89 90-99 100-109 Test Scores

Histograms vs. Bar Graphs Many people confuse histograms with a bar graph. A histogram looks very similar to a bar graph. There are two big differences between a histogram and a bar graph. A bar graph compares items in categories while a histogram displays one category broken down into intervals. For example: A bar graph would compare…the number of apples, to the number of oranges, to the number of bananas at a grocery store. A histogram would compare…the number of people who eat 0-4 apples a week, to the number that eat 5-9, to the number who eat 10-14.

Histograms vs. Bar Graphs The bars on a histogram touch. The bars found on a bar graph do not touch. Why do you think that the bars will touch on a histogram? It will make intervals of data easier to compare.

Skewed to the left Skewed to the right Symmetric

Mean and Median Comparisons If the data is symmetric, the mean and the median are approximately the same. If the data is skewed to the right, the mean is larger than the median. If the data is skewed to the left, the mean is smaller than the median. mean = -0.0373 mean = 10.71 mean = 4.829 median = -0.0173 median = 7.75 median = 6.629

Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincide If a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) Mode Mean Median

Standard Deviation The standard deviation of a set of observations is the square root of the variance . Another measure of where a value x lies in a distribution is its deviation from the mean deviation from the mean = value – mean = x -

Algebra 1-9 Statistics - Analyzing Data by Using Tables and Graphs Some ways a graph can be misleading: Numbers are omitted on an axis, but no break is shown. The tick marks on an axis are not the same distance apart or do not have the same-sized intervals. The percents on a circle graph do not have a sum of 100. Harbour

Misleading Histograms What does it the word “misleading” mean? Deceptive or intentionally create a false impression. Types of Misleading Histograms Combing Intervals: The amount of data in each interval can make a histograms look different. Stretched Graphs: Graphs might be stretched vertically so that data looks larger. Excluded Intervals: Intervals may be skipped on the x or y-axis to make the data look smaller.

Investigating Scatter Plots Scatter plots are similar to line graphs in that each graph uses the horizontal ( x ) axis and vertical ( y ) axis to plot data points. Scatter plots are most often used to show correlations or relationships among data.

Investigating Scatter Plots

Investigating Scatter Plots Positive correlations occur when two variables or values move in the same direction. As the number of hours that you study increases your overall class grade increases

Investigating Scatter Plots – Positive Correlation Study Time Class Grade 55 0.5 61 1 67 1.5 73 2 81 2.5 89 3 91 3.5 93 4 95 4.5 97

Investigating Scatter Plots Negative Correlations occur when variables move in opposite directions As the number of days per month that you exercise increases your actual weight decreases

Investigating Scatter Plots – Negative Correlation Work out time Weight 200 0.5 205 1 190 1.5 195 2 180 2.5 3 170 3.5 177 4 160 4.5 5 150 5.5 168 6 140 6.5 7 130 7.5 8 120 8.5 9 110 9.5 115 10 100 10.5 11 90 11.5 12 80

Investigating Scatter Plots – No Correlation number of shirts owned salary 1 2 3 50 4 30 5 25 6 17 7 8 40 9 10 11 12 13 19 14 55 15 71 16

Line of Best Fit A line of best fit is a line that best represents the data on a scatter plot. A line of best fit may also be called a trend line since it shows us the trend of the data The line may pass through some of the points, none of the points, or all of the points. The purpose of the line of best fit is to show the overall trend or pattern in the data and to allow the reader to make predictions about future trends in the data.

Things to remember A scatter plot with a positive correlation has X and Y values that rise together. A scatter plot with a negative correlation has X values that rise as Y values decrease A scatter plot with no correlation has no visible relationship The line of best fit is the line that best shows the trend of the data

Scatterplots Remember, when looking at scatterplots, look for: Association (or direction) Form Strength Outliers

Strength Strength: At one extreme, the points appear to follow a single stream (whether straight, curved, or bending all over the place). At the other extreme, the points appear as a vague cloud with no discernable trend or pattern. Note: the strength (r).

Form Form: If there is a straight line (linear) relationship, it will appear as a cloud or swarm of points stretched out in a generally consistent, straight form. If the relationship isn’t straight, but curves, while still increasing or decreasing steadily, we can often find ways to make it more nearly straight.

2000 Presidential Election (Outliers)