Math Alliance Project4th Stat Session

Analyzing Quantitative Data

Review

GAISE Statistical problem solving steps:1. formulate a statistical question2. design and implement a plan to collect data3.analyze the data4.interpret the results in the context of the

original question

Types of Data

CategoricalGraphs

Bar GraphsPie Chart

Types of Data

QuantitativeGraphs

Dot plotStemplotHistogramBoxplot

Essential Understanding

• Big Idea 1: The common thread in the statistical problem solving process is the focus on recognizing, summarizing and understanding variability in data. The distribution describes the variability in data.

• There are various ways to represent and summarize a distribution. These include tables, graphs, and numerical summaries. – Representations for the distribution of data on a single

categorical variable include:• Frequency / relative frequency table• Frequency / relative frequency bar graph• Pie chart (circle graph)

Essential Understanding– Representations for the distribution of data on a single numerical

variable include:• Dot plot• Frequency / relative frequency table• Cumulative Frequency / relative frequency table• Stem and leaf plot• Histogram• Box plot

– With numerical data, identify patterns in the variability and describe important features of the distribution including:

• Shape of the distribution– Mound, symmetric, skewed, bi-modal

• Center of the distribution– Mean, Median, mode

• Spread of the distribution– Range, interquartile range, mean absolute deviation, standard deviation

Grade 6 Statistics & Probability

Develop understanding of statistical variability.

• 1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

• 2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

• 3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Grade 6 Statistics & Probability

Summarize and describe distributions.4. Display numerical data in plots on a number line, including dot plots, histograms,

and box plots. 5. Summarize numerical data sets in relation to their context, such as by: • Reporting the number of observations. • Describing the nature of the attribute under investigation, including how it was

measured and its units of measurement. • Giving quantitative measures of center (median and/or mean) and variability

(interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

• Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Essential Understanding

Representations for the distribution of data on a single numerical variable include:

• Dot plot• Frequency / relative frequency table• Cumulative Frequency / relative frequency table

Activity

Statistical Question: How long are the names of students in our class?

Criteria:PopulationMeasurementVariability

GAISE Step 2: collect the data

Complete table

Last Name Length

First Name Length Combined length

Hopfensperger 13 Patrick 7 20

Winn 4 Judy 4 8

Gaise Step 3: Analyze the Data

Dotplot (also called a lineplot)Steps to construct a dotplot:

Horizontal axis scaled to cover the range of the different name – lengthsRead down list and place a (•) or (x) for each length above the appropriate mark

Frequency Table Combined Name – Length Frequency

7 3

8 1

9 0

10 6

11

12

13

14

Reflection

What are the advantages and disadvantages of each type of display? (Dotplot vs. Frequency table)

Does one have to be constructed first in order to construct the other?

Gaise Step 4: Interpret the results

Original question: How long are the names of students in our class?

Write an answer to this question using the dotplot and frequency table to support your findings.

Extension

Have you ever completed a form that does not have enough blanks for your entire name?

Suppose a form has only 15 blanks for the combined name (first, space, last).

How many people in class would be able to enter their entire name?

Cumulative Frequency tableLength Frequency Cumulative Frequency

7 3 3

8 1 4

9 0 4

10 6 10

Relative Cumulative Frequency table

Length Frequency Cumulative Frequency

Relative Cumulative Frequency

7 3 3 .3

8 1 4 .4

9 0 4 .4

10 6 10 1.00

Total 10

Interpret the results

If we wanted to design a form so that “most” of the students in class would have enough room to write their full name, how long should the form be?

Essential Understanding

– With numerical data, identify patterns in the variability and describe important features of the distribution including:

• Shape of the distribution– Mound, symmetric, skewed, bi-modal

Mound, normal, bell-shaped

Symmetric, Uniform

Skewed

Bi-Modal

Stemplot

Example of a stem plotTravel TimesHow many minutes did it take you to get to Gaeslen from your school?Statistical Question?

Burger King Data

Other examples of stemplots

Students and Basketball players Heights (Navigating book p. 88)

Skateboard prices (p. 23 Data Distributions)

Activity: How long is a minute?

How good are you at estimating how long a minute is?

Partner one: Head down and hand upLeave hand up until you think one minute has passed

Partner two:Carefully time and recorded how long your partner kept their hand up

Minute Activity

Switch rolesPartner two – while timing talk to your partner about school, sports, news, your family

Remember to carefully time and record how long your partner has their hand up

Back-to- Back Stemplot

Compare estimating one minute between the groups

What is the statistical question we are attempting to answer?

Construct a back-to-back stemplot to help answer our question.

Histograms

A histogram is a graphical display of a frequency distribution of quantitative data using bars of the same width (class interval) and heights dependent on the frequencies.

How is a Histogram Made?

Consider the set of values:3, 11, 12, 19, 22, 23, 24, 25, 27, 29, 35, 36, 37,

45, 49Construct a frequency table – decide class width

Class Width Tally Frequency

0-10 | 1

10-20 | | | 3

20-30 | | | | | | 6

30-40 | | | | 4

40-50 | | 2

ReflectionWhy are there no gaps between the

bars in a histogram?

Histogram Examples

Burger King Data

Migraines Data p. 94NCTM Navigating through Data Analysis in Grades 6-8

Summary

When would each of these be most useful? What are the advantages and disadvantages of

each type of display?Dotplot

Stemplot

Histogram

Summary

Purpose is to get a visual display of your data and begin to draw some conclusions about the statistical question.

Describe the shapeEstimate center and spread