Statistics: Analyzing Data by Using Tables and Graphs1.8; 1.9; 5.7 & 13.3

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

summer 2004 election

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

Sum of the observationsNumber of observations

Mean =

• This is the most popular and useful measure of central location

The Arithmetic 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

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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

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• Example 1The 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.

00 77 222211.011.0

The Arithmetic Mean

Odd number of observations

0, 0, 5, 7, 8 9, 12, 14, 220, 0, 5, 7, 8, 9, 12, 14, 22, 330, 0, 5, 7, 8, 9, 12, 14, 22, 33

Even number of observations

.

Find the median of the time on the internetfor the 10 adults of example 3.1

• The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude.

The Median

Suppose only 9 adults were sampled (exclude, say, the longest time (33))

Comment

8.5, 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.

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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 n

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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.9.62

2.76.6

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.

Vocabulary

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

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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%.

Circle graph: compares parts of a set of data as a percent of the whole set.

63% Worse8% Better

26% Same

3%Not sure

National Traffic Survey

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.

Line graph: numerical data displayed to show trends or changes over time.

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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

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

Creating a HistogramInterval 50-59 60-69 70-79 80-89 90-99

Frequency of Data 3 2 7 6 6

Test Scores

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Math Test Scores

50-59 60-69 70-79 80-89 90-99 100-109

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Don’t forget little things…like labels

and equal intervals!

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.1. 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

2. 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

Data

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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”)

MeanMedian

Mode

• 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 -

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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.

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 PlotsWeight Loss Over Time

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How Study Time Affects Grades

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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

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How Study Time Affects Grades

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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

Weight Loss Over Time

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Investigating Scatter Plots – No Correlation

How does your wardrobe affect your salary

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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.

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2000 Presidential Election(Outliers)