Displaying Data Visually

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Displaying Data Visually

• Chapter 1.1 The Power of Information
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author G. Greer (with help from K. Myers)

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Why do we collect data?

• one of the ways we learn is to observe
• however, it may be difficult to make accurate
  decisions if we do not observe in an appropriate
  fashion
• collecting data is a systematic method of making
  observations that helps to collect representative
  data and provides others with a way of repeating
  our observations
• good definitions for this chapter at
• http//www.stats.gla.ac.uk/steps/glossary/alphabe
  t.html

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Scales of Measurement of Data

• Nominal Data
• have no order and thus only gives names or labels
  to various categories
• Ordinal Data
• have order, but the interval between measurements
  is not meaningful
• Interval Data
• have meaningful intervals between measurements,
  but there is no true starting point (zero).
• Ratio Data
• have the highest level of measurement. Ratios
  between measurements as well as intervals are
  meaningful because there is a starting point
  (zero).
• (Caulkins, 2003)

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Types of Data

• Discrete Data
• data where a fraction is impossible
• Continuous Data
• data where fractions are possible
• Nominal and Ordinal generally give rise to
  discrete data, while Interval and ratio can give
  rise to either type (James Cook University, n.d.)

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Examples of Scales of Measurement

• Nominal
• data has just a name
• sex, species, marital status, individuals
• Ordinal
• data with order
• Likert scale, level of education
• Interval
• has order and equal interval between levels
• Celsius temperature, tide height, longitude
• Ratio
• Kelvin Temperature, length, weight
• order, equal interval, absolute starting point

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Who do we collect data from?

• A population is the entire group from which we
  can collect data
• A census occurs when we collect data from all the
  members of the population
• However, this may take too long or be too
  expensive
• Thus we often take a sample of the population,
  and work on the assumption that this is
  representative of the entire population
• One of the large questions is whether the sample
  represents the population. We will come back to
  this in a later chapter.

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Organizing Data

• A frequency table is often used, listing the
  variable and the frequency.
• What type of data does this table contain?
• intervals cant overlap
• use 3-12 intervals
• can also be done without intervals

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Organizing Data

• Another useful organizer is a stem and leaf plot.
  
• What type of data is this?
• The class interval is the size of the grouping,
  and is 10 units here.
• What is the effect of reducing or increasing the
  interval size?
• stem can have as many numbers as needed
• each leaf must be recorded for each time the
  number occurs

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Measures of Central Tendency

• used to indicate the one value that best
  represents a group of values
• arithmetic mean
• add all numbers and divide by the count of values
• median
• place all values in order and chose middle number
• less affected by outliers
• mode
• most common number
• may be none, one or many modes

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Displaying Data Bar Graphs

• Typically used for nominal/discrete data.
• Why are the bars separated?
• Would it be incorrect if you didnt separate
  them?

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Displaying Data - Histograms

• Typically used in continuous data (interval and
  ratio)
• Why are the bars attached? (because the x-axis
  represents intervals)
• Choice of class interval size is important. Why?

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An example

• these are prices for Internet service packages
• find the mean, median and mode
• determine what type of data this is
• create a suitable frequency table, stem and leaf
  plot and graph
• 13.60 15.60 17.20 16.00 17.50 18.60 18.70
• 12.20 18.60 15.70 15.30 13.00 16.40 14.30
• 18.10 18.60 17.60 18.40 19.30 15.60 17.10
• 18.30 15.20 15.70 17.20 18.10 18.40 12.00
• 16.40 15.60

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Answers

• Mean 494.30/30 16.50
• Median average of 15th and 16th numbers
• Median (16.40 17.10)/2 16.75
• Mode 15.60 and 18.60
• What type of data? numerical, so at least
  Interval data. It has an absolute starting
  point, so it is ratio data
• Given this, a histogram is appropriate

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Displaying Data Pie Graphs

• a circle divided up to represent the data
• see page 8 of the text for an example of creating
  these by hand

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Examining the spread of data

• the box and whisker plot is one way to indicate
  the spread of data
• why is this important?
• instructions for creating these may be found on
  page 9 of the text or at http//regentsprep.org/R
  egents/math/data/boxwhisk.htm

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Examining Trends

• the obvious way of seeing trends is to create a
  broken-line graph

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Conclusions and Issues in One Variable Data

• Chapter 1.2 The Power of Information
• Mathematics of Data Management (Nelson)
• MDM 4U
• Author G. Greer (with ideas from K. Myers)

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What conclusions are possible?

• to draw a conclusion, a number of conditions must
  apply
• data need to be representative
• sample size must be large enough
• data must address the question
• good definitions for this chapter at
• http//www.stats.gla.ac.uk/steps/glossary/alphabe
  t.html

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Drawing Conclusions

• do females seem more likely to be interested in
  student government?
• does gender appear to have an effect on interest
  in student government?
• is this a correlation?
• is it likely that being female causes interest?

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What does a relationship mean?

• what is a correlation?
• a correlation occurs when two variables appear to
  be linked
• that means that you have evidence that a change
  in one variable is associated with a change in
  the other
• ex amount of study with increasing day length in
  January
• what is a causal relationship?
• this is a relationship where a change in one
  variable can be proved to cause a change in the
  other
• usually requires an in-depth study to prove
• ex amount of study with the approach of exams

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Exercises

• try p. 11 2, 3a, 3b, 4, 7, 8
• try p. 21 4, 9, 11, 14

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Investigation

• we will analyze data of male and female heights,
  looking to display the data in a meaningful way
  and draw some conclusions from the data
• in the exercise we will use a spreadsheet and
  Fathom, as students are expected to be familiar
  with both of these
• the investigation is available at
• http//napaneedss.limestone.on.ca/greer/mdm4u/chap
  ter1/investigation1.pdf

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Investigation

• gender nominal
• height ratio
• mean/median/mode
• what is the population?
• how was the data collected?
• what conclusions?

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References

• Calkins, K. (2003). Definitions, Uses, Data
  Types, and Levels of Measurement. Retrieved
  August 23, 2004 from http//www.andrews.edu/calki
  ns/math/webtexts/stat01.htm
• James Cook University (n.d.). ICU Studies Online.
  Retrieved August 23, 2004 from
  http//www.jcu.edu.au/studying/services/studyskill
  s/scientific/data.html