Topic 3:Introduction to

Statistics

Algebra 1

Table of Contents1. Introduction to Statistics & Data2. Graphical Displays3. Two-Way Tables4. Describing Distributions: Shape, Skew &

Center5. Measures of Spread

What is the Study of Statistics?!

• Statistics is the science of data.

• Statistics is the mathematic discipline that involves collecting and analyzing data.

Collecting Data• We collect data through observation, surveys

and experiments. • We can collect two different types of data:

– Categorical– Quantitative

Data Collected

Categorical Variable- Usually an adjective- Rarely a numberExamples:- Gender- Race- Grade in School

(Freshmen, Soph, Jr., Sr.)

- Zip Code

Quantitative Variable - Always a number- Must be able to find the

mean of the numbersExamples:- Weight- Height- Amount of money in

wallet- Age

Categorical or Quantitative?1. Survey about whether student buy lunch from the cafeteria or bring lunch from home, doesn’t eat lunch, etc.

2. Experiment where we measure how tall a plant grows.

3. Observation where we count how many people are in each car leaving school.

4. Survey about each student’s shoe size.

Graphical Displays of Data

Displaying Data• We can display data

in a variety of ways.• Based on the type of

data collected (categorical or quantitative) and the amount of data we select the best style of graph.

Displaying Categorical DataPie Chart: Bar Graph:

Displaying Quantitative Data

MPG14 16 18 20 22 24 26 28 30 32 34

2009 Fuel Economy Guide Dot PlotDotplot

Histogram

Boxplot

Dotplots– Each data value is shown as a dot above its

location on a number line.Number of Goals Scored Per Game by the 2004 US Women’s Soccer

Team

3 0 2 7 8 2 4 3 5 1 1 4 5 3 1 1 3

3 3 2 1 2 2 2 4 3 5 6 1 5 5 1 1 5

1. Draw a horizontal axis (a number line) and label it with the variable name.

2. Scale the axis from the minimum to the maximum value.

3. Mark a dot above the location on the horizontal axis corresponding to each data value.

How to Make a Dotplot

Let’s Practice…

Histograms– Looks like a bar graph, but the bars must

touch!

– X-axis labeled with number ranges

1) Divide the range of data into classes of equal sizes.

2) Find the count (frequency) of individuals in each class.

3) Label and scale your axes and draw the histogram. The height of the bar equals its frequency. Adjacent bars should touch, unless a class contains no individuals.

How to Make a Histogram

Let’s Practice…

Box and Whisker PlotsA box plot is a graphical display of the minimum, first quartile, median, third quartile, and maximum.

The term "box plot" comes from the fact that the graph looks like a rectangle with lines extending from the top and bottom.

Quartiles• We can divide data in quartiles• Quartiles are divisions representing 25% of the

data.

How to Calculate QuartilesTo calculate the quartiles:1)Arrange the observations in increasing order and

locate the median M.2)The first quartile Q1 is the median of the

observations located to the left of the median in the ordered list.

3)The third quartile Q3 is the median of the observations located to the right of the median in the ordered list.

Let’s Practice…Calculate the 1st, 2nd (median) and 3rd quartiles for the following data sets:

1. 15, 17, 16, 15, 18, 19, 15, 20, 18

2. 5, 8, 9, 7, 6, 9, 8 ,7, 10, 11, 4

Interquartile Range (IQR)

Let’s Practice: Find the Quartiles and calculate the IQR.

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Travel times to work for 20 randomly selected Miami Residents:

5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Travel times to work for 20 randomly selected Miami Residents

5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85

M = 22.5 Q3= 42.5Q1 = 15

IQR = Q3 – Q1

= 42.5 – 15= 27.5 minutes

Simple Box & Whisker Plot

How to Make a Simple Box & Whisker Plot

1. Draw a number line.2. Mark the median, Quartile 1 and Quartile 3.3. Draw a box around Q1 and Q3.4. Mark the lowest and highest values with a

dot.5. Draw whiskers from the end of each box to

the dot.

Let’s Practice: Create a Box and Whisker Plot.

25 30 26 30 29 26 22 23 24 23 25 28Quiz Scores:

Modified Box & Whisker Plot

• Modified Box & Whisker plots highlight outliers.

• Outliers are extreme values. • Can be much higher or lower than the rest of

the data.

Outliers

How to Determine Outliers1. Calculate IQR.2. Calculate lower fence

Q1 – (1.5 * IQR)3. Calculate upper fence.

Q3 + (1.5 * IQR) 4. Outliers are any values outside of the fences.

How to Make a Modified Box & Whisker Plot

1. Draw a number line.2. Mark the median, Quartile 1 and Quartile 3.3. Draw a box around Q1 and Q3.4. Find outliers5. Mark the lowest and highest non-outlier

values with a dot.6. Draw whiskers from the end of each box to

the dot.7. Draw a dot for the outliers.

Let’s Practice: Create a Modified Box & Whisker Plot.

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

Travel times to work for 20 randomly selected Miami Residents:

Let’s Practice:

TravelTime0 10 20 30 40 50 60 70 80 90

Collection 5 Box Plot

Let’s Practice: Construct a Modified Box & Whisker Plot

M = 22.5 Q3= 42.5Q1 = 15Min=5

10 30 5 25 40 20 10 15 30 20 15 20 85 15 65 15 60 60 40 45

5 10 10 15 15 15 15 20 20 20 25 30 30 40 40 45 60 60 65 85

Let’s Practice…Identify the median, Q1 and Q3, and the IQR.1.

2.

Let’s Practice…Identify the median, Q1 and Q3, and the IQR. If you had to pick one career, which one would you pick and why? (Must be a statistical reason!)

In tens of thousands of dollars.

Name the Type of Graph

Name the Type of Graph

Name the Type of Graph

Name the Type of Graph

Name the Type of Graph

Two-Way Tables

Two-Way Tables Two-Way Tables: describe two categorical variables, organizing counts according to a row variable and a column variable.When a dataset involves two categorical variables, we begin by examining the counts or percents in various categories for one of the variables.

Let’s Practice…

1. What proportion of students have red hair?2. What proportion of students have brown eyes and

hair?3. What proportion of students have blue eyes and

either red or blond hair?

Let’s Practice…

4. What proportion of students have not brown eyes and black hair? 5. What proportion of students with blond hair have blue eyes?6. What proportion of students with hazel eyes have a hair color other than brown?

Let’s Practice…1. What proportion of students that ride the school bus are members of two or more clubs?2. What proportion of students that are members of no clubs do not ride the school bus?3. What proportion of students that do not ride the school bus are members of at least one club?

Member of No Clubs

Member of One Club

Member of 2 or More Clubs Total

Rides the School Bus 55 33 20 108

Does not Ride Bus 16 44 82 142

Total 71 77 102 250

Describing Distributions:

Shape, Skew & Center

Different Shapes of Distributions

• Distributions can be described as:– Roughly symmetric– Skewed right– Skewed left

Shape Definitions:Symmetric: if the right and left sides of the graph are approximately mirror images of each other.

Skewed to the right (right-skewed) if the right side of the graph is much longer than the left side.

Skewed to the left (left-skewed) if the left side of the graph is much longer than the right side.

DiceRolls0 2 4 6 8 10 12

Collection 1 Dot Plot

Score70 75 80 85 90 95 100

Collection 1 Dot Plot

Siblings0 1 2 3 4 5 6 7

Collection 1 Dot Plot

Symmetric Skewed-left Skewed-right

Skew in Box Plots

Describe the Shape…

Describe the Shape…

Other Ways to Describe Shape:

• Unimodal

• Bimodal

• Multimodal

Measures of Center• Measures of Center = Mean and Median

Type of Distribution Best Measure of CenterSymmetric MeanSkewed Right MedianSkewed Left Median

Why?!?!

Which Measure of Center?

Measures of Spread

Standard Deviation, IQR and Range

Standard DeviationStandard deviation is a number used to tell how measurements for a group are spread out from

the mean.

• A relatively low standard deviation value indicates that the data points tend to be very close to the mean.

• A relatively high standard deviation value indicates that the data points are spread out over a large range of values.

Standard Deviation

Below are dotplots of three different distributions, A, B, and C. Which one has the

largest standard deviation? Justify your answer.

Measures of Spread• Measures of Center = IQR, Range and

Standard Deviation

Type of Distribution Best Measure of CenterSymmetric Standard Deviation

Range

Skewed Right IQRSkewed Left IQR

Which Measure of Spread?

Let’s Practice..Mr. Morris gave his algebra class a test, the results of which are listed below. 68, 92, 74, 75, 86, 90, 92, 81, 60, 82, 77, 80 Shania was absent on the day of the test and had to take the test late. She earned a score of 99. Which measure of the class's test results did Shania's score most change? A. IQR B. Mean C. Median D. Range