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Basic Statistical Tools

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Agenda

  • Simple Data Tools
  • Dangers in Simplification
  • Examples

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Simple Data Tools

How Can We Quantify What We See?

An Overview

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Challenges With Statistics

  • What are the challenges in statistics?
  • Discuss with partners

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Challenges With Statistics

  • What are the challenges in statistics?
  • Discuss with partners
    • Data can be “dirty” - errors, bias, measurement
    • There are often unmeasurable influences on measurable variables
    • The measured variables aren’t necessarily what you want to measure
    • The mathematics is relatively easy - the interpretation is always hard

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Statistics Process

  • Collect data
  • Organize it
  • Analyze it
  • Represent it
  • Interpret and then reflect on conclusions

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Example Data

  • Data for student marks on a quiz
  • Located here
  • Follow the process
    • Collect (done), Organize, Analyze, Represent, Interpret and Reflect
    • For the Analyze step, find the centre
    • How spread out is the centre?

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Example Data

  • Data for student marks on a quiz
  • Centre:
    • Mean - 5.25
    • Median - 5.5
    • Mode - 7
  • Spread of data:
    • Variance - 6.39
    • Standard deviation - 2.53

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Finding the Centre: Mean, Median, Mode

  • What does it mean to be average?

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Finding the Centre: Mean, Median, Mode

  • What does it mean to be average?
  • Approximately the “middle” of the data
    • Mathematically, this is often called the centre
  • Some data has a centre that can be interpreted
  • Other data has a meaningless centre
    • Bi-modal, skewed, multi-modal, exponential distribution

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Finding the Centre: Mean, Median, Mode

  • Mode
    • Most common element
  • Median
    • Exactly half elements less and half more than the value
  • Mean (average)
    • Sum of elements divided by number of elements
    • ​

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Mean

  • Example 1) Find the mean grade (%): 65, 62, 75, 80, 71, 15

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Weighted Mean

  • Example 2: This table shows the ages of students attending a university application workshop. What is the mean age?

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Age

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Number of people

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6

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Weighted Mean

Example 3: A teacher assigns the following weights:

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A student achieves marks of 70%, 84% and 82% in the first 3 categories respectively. What is the student’s mark going into the exam?

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What mark do they need on the final exam to earn a 78% in the course?

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Grouped Data

The following table shows recent exam results and how many students scored in each interval. Find the mean grade.

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Median

Median – the middle data point when the data is listed in numerical order. If there is an even number of elements, use the mean of the middle two points.

Example 5: Find the median grade of the dataset used earlier:

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65, 62, 75, 80, 71, 15

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How does this differ from the mean? Why?

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Mode

Mode – the most frequently appearing element.

  • If there are no repeating elements, there is no mode.
  • If two or more elements occur most often, they are called bimodal (2), trimodal (3) or multimodal (4 or more).
  • This is the only measure suitable for non-numeric (qualitative) data.

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Choosing the appropriate measure:

  • Mean is the most widely used and reported
  • Median used if:
    • Range of data is open ended (e.g. “x or greater” is a bin)
    • Dataset contains a few large outliers
    • Dataset is skewed
  • Mode is used for nominal/categorical data (data that is not ordered)

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From the QuestionBank:

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From the QuestionBank:

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From the QuestionBank:

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From the QuestionBank:

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From the QuestionBank:

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  1. What is the value of ‘p’?

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From the QuestionBank:

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  • What is the value of ‘p’?

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From the QuestionBank:

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b) What is the modal class?

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From the QuestionBank:

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b) What is the modal class?

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From the QuestionBank:

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c) Approximately 50% of performances sold less than ‘a’ tickets. Find ‘a’.

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From the QuestionBank:

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c) Approximately 50% of performances sold less than ‘a’ tickets. Find ‘a’.

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Measures of centre

with technology…

In Excel/Google Sheets: (same formulas work in both applications)

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Mean: =average(range)

Median: =median(range)

Mode: = mode(range)

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Weighted mean:

=SUMPRODUCT(range1,range2)/sum(range2)

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Measures of centre

with technology…

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Measures of centre

with technology…

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Desmos

AP Stats 1.4

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Width of the Centre: Variance, Standard Deviation

  • Width of the centre
  • How the data spreads out from the centre
  • Only makes sense with normal distributions
    • Also called Gaussian distributions
  • Variance

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  • Standard deviation - square root of variance

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Finding Standard Deviation

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65, 62, 75, 80, 71, 15

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Mark Distance from the mean (Distance from the mean)2

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Width of the Centre: Variance, Standard Deviation

  • Standard deviation - square root of variance
  • “Most” of the data is clustered around the mean
  • Distribution has long tails

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https://commons.wikimedia.org/wiki/File:Standard_deviation_diagram.svg#/media/File:Standard_deviation_diagram.svg

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Standard Deviation for Ungrouped Data on the GDC

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Standard Deviation For Grouped Data on the GDC

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Fitting Curves to Data

  • Approximating the data as a single curve
    • For single variable data
  • There are many ways to fit curves to data
  • A popular method is called Least Squares
    • Minimizes (using Calculus) the sum of the squares of the difference between the curve and the data points
  • Each method has trade offs - there is no single “best” method
  • More later in the course!

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Dangers in Simplification

What Can Go Wrong With Losing Information?

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Golden Rules of Statistics

  • Correlation is not causation
  • Simplification removes information, decreases subtlety and nuance
    • Corollary: second order effects can reverse conclusions
  • Averages of averages are meaningless
  • Data can be “dirty”
    • Bias, error, omissions

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Example: Correlation is not Causation, Ex 1

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Example: Correlation is not Causation, Ex 2

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Example: Simplification Removes Information

  • What is going on here?

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https://www.autodeskresearch.com/publications/samestats

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Qualitative and Quantitative Data

  • Qualitative:
    • Categories of data
    • More descriptive
    • What is your favourite pen colour?
    • How do you travel to school?
  • Quantitative:
    • Information that can be counted or measured
    • How many pens do you own?
    • How long did it take to get to school today?

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Examples

Putting It Into Practice

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Example 1

  • Describe the skew for the data shown:

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Example 1

  • Describe the skew for the data shown:

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  • Right Skew, Normal, Left Skew
  • Only the normal distribution has a sensible measure of central tendency (averages)

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Example 2

  • A group of students was surveyed to find out how many hats each of them owned
  • Graph and comment on the data and central tendency

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Hats

Freq

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3

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7

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3

3

3

4

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Example 2

  • A group of students was surveyed to find out how many hats each of them owned
  • Graph and comment on the data and central tendency
  • 18 students surveyed
  • Mode: 1
  • Median: 1
  • Mean: 1.67
  • Std Dev: 1.28
  • Hat Data

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Hats

Freq

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3

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7

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Example 3

  • All of the IB students in the school were asked how many minutes a day they spent on mathematics outside of class
  • Is the data continuous or discrete?
  • Is it qualitative or quantitative?
  • Is it normal or skewed?
  • Visualize and then infer meaning
  • How to calculate mean, median, and mode in this case?

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Time (min)

# Students

0 - 15

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15 - 30

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30 - 45

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45 - 60

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60 - 75

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75 - 90

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Example 3

  • Discrete data
    • Quantitative?
    • More or less normal data
    • Slight skew left
  • How to calculate mean, median, and mode in this case?
    • Assume value in the centre of each bin, then use normal calculations
    • In this case, the values are 7.5, 22.5, 37.5, etc.

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Time (min)

# Students

0 - 15

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15 - 30

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30 - 45

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45 - 60

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60 - 75

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75 - 90

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Example 4

  • Our Measurements:
  • Most students in our class are fully grown (height)
    • Let’s find out!
  • Measure height and analyze

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Example 5

  • How many times do you smile each day?
    • Write down a guess
  • How many times does an average adult smile each day?
    • Write down a guess
  • How many times does an average child smile each day?
    • Write down a guess

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Example 5

  • How many times do you smile each day?
    • Write down a guess
  • How many times does an average adult smile each day?
    • Write down a guess
  • How many times does an average child smile each day?
    • Write down a guess
  • Guesses from class

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Example 5

  • How many smiles would you expect from this class in one day?
  • What is the “middle”, and spread of data?
  • Add a child to the class
    • How does this change anything?

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Example 5

  • How many smiles would you expect from this class in one day?
  • What is the “middle”, and spread of data?
  • Add a child to the class
    • How does this change anything?

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Potentially dubious reference on smiling

Happy adult - 40-50, “Normal” adult - 20, Child - 400

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CREDITS

Special thanks to all the people who made and released these awesome resources for free:

  • Presentation template by SlidesCarnival

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