definitions data: a collection of information in context. population: a set of individuals that we...

Definitions

• Data: A collection of information in context.• Population: A set of individuals that we wish

to describe and/or make predictions about.• Individual: Member of a population.• Variable: Characteristic recorded about each

individual in a data set.

DAY 1

Types of Data

Categorical – places individuals into groups (sometimes referred to as qualitative)

Example: gender, eye color, zip code, dominant hand

Quantitative – consists of numerical values (it makes sense to find an average)

Example: height, weight, income, vertical leap

Graphs for Categorical data are Bar graphs and Pie charts

Be sure to label the graph and use proper scales.

Example 1: Reading and Interpreting Bar Graphs

Use the graph to answer each question.

A. Which casserole was ordered the most?

B. About how many total orders were placed?

C. About how many more tuna noodle casseroles were ordered than king ranch casseroles?

D. About what percent of the total orders were for baked ziti?

Check It Out! Example 1


a. Which ingredient contains the least amount of fat?

b. Which ingredients contain at least 8 grams of fat?

Example 2: Reading and Interpreting Double Bar Graphs


A. Which feature received the same satisfaction rating for each SUV?

B. Which SUV received a better rating for mileage?

A double-bar graph ______________________________________________________________________________________________________________________________________


Use the graph to determine which years had the same average basketball attendance. What was the average attendance for those years?

Tips for Making a Bar Graph

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•

•

•

Making a Bar GraphHere are the distributions for car colors in North America in 2008.

Color Percent of VehicleWhite 20Black 17Silver 17Blue 13Gray 12Red 11Beige/Brown 5Green 3Yellow/Gold 2

a.) What percent of vehicles had colors others than those listed?

b.) Display the data in a bar graph.

c.) Would it be appropriate to make a pie chart? If not explain why. If it is appropriate for this data then create one.

Tips for Making the Pie (or Circle) Graph

• You must convert the __________________by multiply the percent value by_______.

• Draw a radius somewhere on the circle and start there and work your way around the circle counter-clockwise until you get back to the radius you started with.

Color Percent of VehicleWhite 20Black 17Silver 17Blue 13Gray 12Red 11Beige/Brown 5Green 3Yellow/Gold 2

Measuring CentersMean (“average”) –

X = X1 + X2 + X3 + … + Xn

nSum of observationsNumber of observations=

Median –

1.) Place data in numerical order2.) If there are an odd number of observations, the median is the center observation in the ordered list.3.) If there are an even number of observations, the median is the average of the two center observations in the ordered list.

Mode –

It helps if you put the data into numerical order. (STAT, edit, enter numbers, STAT, sortA(, enter, 2nd (1 or 2), enter. Go back to list and the numbers will be in order.

If the Mean, Median and Mode for the following observations

78, 98, 48, 63, 84, 100, 95, 86, 91 ,87, 48, 94, 94, 89 ,95, 95, 97, 41 ,65, 85

Mean = 78+98+48+63+84+100+95+86+91+89+48+94+94+87+95+95+97+41+65+8520

= 81.65

Median: 41, 48, 48, 63, 65, 78, 84, 85, 86, 87, 89, 91, 94, 94, 95, 95, 95, 97, 98, 100

median is 87+89 = 88 2

Mode is 95

Use the data set below to complete each table.12, 4, 8, 15, 7, 11, 9, 5, 4, 5, 8, 5, 4, 4, 3, 5, 4, 1, 3, 6, 8, 7, 4, 3, 2, 0, 1, 3, 7, 9, 9, 5, 14, 9

Measure of Center

Brief Definition Value

Mean

Median

Mode

• 1. The weights in pounds of six members of a basketball team are 161, 156, 150, 156, 150, and 163. Determine the mean, median, mode and range of the data set.

End of Day 1

Graphs for Quantitative Data• Dotplots –

• Box and Whisker –

Dotplots

• A dotplot is a graph where dots are used to represent individual data points.

• The dots are plotted above a number line. • Dotplots can be used to represent frequencies

for categorical or quantitative data.• Dotplots can be used to see how data items

compare.

Draw a DotplotDraw a dotplot for the data set below.

25, 25, 20, 25, 16, 20, 25, 30, 25, 31, 26, 28, 30

Box and Whiskers

Finding First and Third QuartileThe first quartile lies one quarter of the way up the list. (25% of the data is below the first quartile.)

The third quartile lies three quarters of the way up the list. (75% of the data is below the third quartile.)

1.) Find the median to divide the data is half.2.) Find the “median” of the lower half, this point will be the first quartile.3.) Find the “median” of the upper half, this point will be the third quartile.

***In the case of two data points being either the first or third quartiles, then use same method as the median (add together and divide by 2)

Measures of Spread

Range –

Interquartile Range (IQR)

How much do values typically vary from the center?

2. Use the data to make a box-and-whisker plot.

13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23

Find the minimum, maximum, median (Q2), lower quartile (Q1), and upper quartile (Q3) for the following sets of data and draw the Box and

Whisker plot (or Boxplot)1) 32, 40, 35, 29, 14, 32

2) 6, 1, 7, 6, 5, 5, 0, 1, 0, 8, 4

3) 121, 143, 98, 144, 165, 118

Checking for OutliersAn outlier is data point that is extremely far

away from the rest of the data and may effect some of the measurements we take from that

data.

An outlier is any point that is farther than 1.5 x the IQR from the first or third quartile.

Now, check the previous data for outliners.

Find any outliers, if any. 1) 32, 40, 35, 29, 14, 32

2) 6, 1, 7, 6, 5, 5, 0, 1, 0, 8, 4

3) 121, 143, 98, 144, 165, 118

Examples

Identify the outlier in the data set {16, 23, 21, 18, 75, 21} and determine how the outlier affects the mean, median, mode, and range of the data. outlier: _________________

with the outlier without the outlier

End of Day 2

Graphs for Quantitative DataStem and Leaf Plots – The stem is the larger place value and the leaves are the smaller place values. This graph is used to give a description of the distribution while using the actual values. Important to have a key for the reader.

Histograms –

DAY 2

Example 1A: Making a Stem-and-Leaf Plot

The numbers of defective widgets in batches of 1000 are given below. Use the data to make a stem-and-leaf plot.

14, 12, 8, 9, 13, 20, 15, 9, 21, 8, 13, 19

Number of DefectiveWidgets per Batch

Stem Leaves0 8 8 9 91 2 3 3 4 5 92 0 1

The tens digits are the _________.

The ones digits are the _________.

List the leaves from ___________ to

_____________ within each row.

Title the graph and add a key.Key: 1|9 means 19

Team A: 65, 42, 56, 49, 58, 42, 61, 55, 45, 72

Team B: 57, 60, 48, 49, 52, 61, 58, 37, 63, 48

Team A Team B

3 4

5 6

7

The _________ digits are the stems.

The _________ digits are the leaves.

Title the graph and add a key.

Key:

Team A’s scores are on the left sideand Team B’s scores are on the right.

The season’s scores for the football teams going to the state championship are given below. Use the data to make a back-to-back stem-and-leaf plot.

Example 1B: Making a Stem-and-Leaf Plot


The temperature in degrees Celsius for two weeks are given below. Use the data to make a stem-and-leaf plot.

7, 32, 34, 31, 26, 27, 23, 19, 22, 29, 30, 36, 35, 31

Stem Leaves

The frequency of a data value is ___________________

______________________________________________

______________________________________________

A frequency table ______________________________

______________________________________________

______________________________________________

______________________________________________

Example 2: Making a Frequency Table The numbers of students enrolled in Western Civilization classes at a university are given below. Use the data to make a frequency table with intervals.

12, 22, 18, 9, 25, 31, 28, 19, 22, 27, 32, 14

Step 1: ___________________________________

Step 2: ____________________________________________________

Example 2 Continued

Number Enrolled

Frequency

1-10 1

11-20 4

21-30 5

31-40 2

Enrollment in WesternCivilization Classes

Step 3: ________________________________________

_______________________________________________

_______________________________________________


The number of days of Maria’s last 15 vacations are listed below. Use the data to make a frequency table with intervals.

4, 8, 6, 7, 5, 4, 10, 6, 7, 14, 12, 8, 10, 15, 12

Step 1: Identify the least and greatest values.

Step 2: Divide the data into equal intervals.

Step 3: List the intervals in the first column of the table. Count the number of data values in each interval and list the count in the last column. Give the table a title.

Check It Out! Example 2 Continued

Interval Frequency

What Is a Histogram?

• A histogram is a bar graph that shows the distribution of data.

• A histogram is a bar graph that represents a frequency table.

• The horizontal axis represents the intervals. • The vertical axis represents the frequency. • The bars in a histogram have the same width

and are drawn next to each other with no gaps.

Constructing a HistogramStep 1 - Count number of data pointsStep 2 - Compute the rangeStep 3 - Determine number of intervals (5-12)Step 4 - Compute interval widthStep 5 - Determine interval starting & ending

pointsStep 6 - Summarize data on a frequency tableStep 7 - Graph the data

Example 3: Making a Histogram

Use the frequency table in Example 2 to make a histogram.

Step 1: _____________________

____________________________

____________________________

Number Enrolled

Frequency

1 – 10 1

11 – 20 4

21 – 30 5

31 – 40 2

Enrollment in WesternCivilization Classes

A histogram is __________________________________________

_______________________________________________________

_______________________________________________________

_______________________________________________________

Example 3 Continued

Step 3: _______________________________________________

Step 2: ______________________________________________

_____________________________________________________

All bars should be the same width. The bars should touch, but not overlap.


Make a histogram for the number of days of Maria’s last 15 vacations.

4, 8, 6, 7, 5, 4, 10, 6, 7, 14, 12, 8, 10, 15, 12

Interval Frequency

4 – 6 5

7 – 9 4

10 – 12 4

13 – 15 2

Number of Vacation Days

Step 1: Use the scale and interval from the frequency table.

The data below shows the number of hours per week spent playing sports by a group of students.

1) What is the minimum, maximum, & range?2) Make a frequency tables using intervals you decide on.3) Draw a histogram.

2 7 17 9 6 13 8 4

5 12 3 11 1 8 15 6

0 3 6 9 12 15 18

1

2

3

4

5

Hours per week playing sports

Freq

uenc

y

End of Day 3Quiz tomorrow

Describing Distributions

•Center •Shape

•Spread

•Outliers

Center

For describing the center of the data, we can use either the Median or the Mean.

Both are useful, in cases where the data is skewed one way the Median is a better choice because it is more resistant to outliers.

Shape•“Symmetrical/Normal”

•“Skewed Left”

•“Skewed Right”

Spread

Range - is the difference of the maximum and minimum value - spread of the entire data set

Interquartile Range (IQR) - is the difference of the upper quartile (Q3) & the lower quartile (Q1) – spread of the middle 50% of the data

We can also use the Standard deviation

Even though we are not graphing a Box and whisker we still use these two measures of spread.

Standard Deviation• is another measure of spread

• is the typical amount that a data value will vary from the mean

• the larger the standard deviation, the ______ ________________the data set (the data points are far from the mean )

• the smaller the standard deviation, the ______ _______________ the data set (the data points are clustered closely around the mean.)

DAY 6

Calculating Standard Deviation

Step 1 –Step 2 –Step 3 –Step 4 –Step 5 –Step 6 –

Standard DeviationFind the standard deviation for this set of data.

3, 5, 5, 7, 8, 9, 9, 10

mean =variance =

standard =deviation

Value Deviation from mean

Squared Deviation

3557899

10

3 – 7 =-4 5 – 7 = -2 75 – 7 = -27 – 7 = 08 – 7 = 19 – 7 = 29 – 7 = 210 – 7 = 3

16 440 16+4+4+0+1+4+4+9 = 42 = 6 1 (8 – 1) 7 449

6 = 2.45

Find the Standard Deviation for the Following Data

Example #1 1, 3, 4, 4, 4, 5, 7, 8, 9

Example #2338, 318, 353, 313, 318, 326, 307, 317

Remember to find the mean first.

Standard deviation = 2.55

Standard deviation = 14.99

• The owner of the Ches Tahoe restaurant is interested in how much people spend at the restaurant. He examines 10 randomly selected receipts for parties of four and writes down the following data. Find the standard deviation.

44, 50, 38, 96, 42, 47, 40, 39, 46, 50

• Practice Problem #1: Calculate the standard deviation of the following test data by hand. Use the chart below to record the steps.

Test Scores: 22, 99, 102, 33, 57, 75, 100, 81, 62, 29

Standard DeviationFind the standard deviation for this set of data.

2, 3, 5, 5, 7, 8, 9, 9, 10, 12

mean =

variance =

standard =deviation


Deviation Squared

23557899

1012

Practice!Find the standard deviation for this set of data.

6, 12, 4, 13, 7, 12, 11, 5

mean =

variance =

standard =deviation


Deviation Squared

6124

137

12115

End of Day 5Project tomorrow

definitions data: a collection of information in context. population: a set of individuals that we...

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