Week 3 - Visualizing and Describing Data

Questions to Ask About Data

• Frequency Distribution: How often does each score appear?

  • Shape: Do the scores center around one value?

• Central Tendency: What is a typical score? Give me a single-number summary

• Variability: How well does that single number represent all scores? Are more scores close to it, or are they spread out across a wide range?

1. Frequency Distribution

• How often does each score appear?

• Categorical data:

  • Are there equal number of people with blue, brown, green, and hazel eyes? Or are there some eye colors that are more common? By how much?

• Continuous data:

  • Are there equal number of very unhappy, mildly unhappy, and very happy people? Which happiness level has more people? How many more?

• How are responses distributed across levels?

Distribution: Categorical Data

• Frequency Tables: the tally of how often each category (value) occurs in the data

• Bar Charts: frequency of each category expressed as the height of a bar

Distribution: Continuous Data

• Histogram: frequency of each range of values expressed as the height of a bar

  • Bin size

    • some data; some details get lost depending on the width of the bins (bars)

  • Shape

    • number of peaks, symmetry, tail

    • a normal distribution: a unimodal (one peak), symmetric bell curve with a particular shape

Shape of Distributions: Number of Peaks

• multimodal distributions that have more than one peak need multiple summaries

  • there is not one “overall” tendency you can draw from it

Shape of Distributions: Symmetry/Skew

Summarize Data with Descriptive Statistics

• Descriptive Statistics: concise summaries to describe your data

  • central tendency: what is typical? give me a single number summary

  • variability: how spread out are the scores?

    • How well does that single number represent all scores?

      • are most scores close to it, or are they spread out across a wide range?

Central Tendency: What is typical?

1. Mean

2. Median

3. Mode

Mean: how to calculate

• add all the scores together and divide by the total number of scores

• mean = sum of all scores / number of scores

On a quiz with 50 students, all scores remained the same from quiz 1 to quiz 2, except the highest scorer improved from 6 to 10. Which measure of central tendency changed?

Mean

In a positively skewed distribution, fewer data fall above the mean.

True

Which Central Tendency Measures Should I Use?

• Mean accounts for all scores, but outliers swing the mean a lot. NOT robust

• If skewed or have outliers, use Median

• For a unimodal distribution,

  • Symmetric: Mode = Median = Mean

  • Negative Skew

    • fewer data fall below the mean

  • Positive Skew: Mode < Median < Mean

    • fewer data fall above the mean

Variability: How spread out are the scores?

1. Range

2. Interquartile Range (IQR)

3. Variance

4. Standard Deviation (SD)

Range and Interquartile Range (IQR)

• Range: Highest score - lowest score

  • provides some senes of how far apart the scores are

  • vulnerable to extreme scores because it only depends on two scores (highest and lowest)

  • commonly reported for participant age (shows no one is under 18)

• Interquartile Range (IQR): Top 25% - Bottom 25% (Middle 50%)

  • provides some sense of how representative the median is

  • robust to extreme scores

Median and IQR

Box Plots: Visualizing Median, IQR, and Range

Box Plots Show Skewness

How much do the scores deviate from the Mean?

Sum of Squares, Variance, and Standard Deviation

Example: SS, Variance, and SD

Population Parameter vs. Sample Statistic

Fixing the Underestimated Sample Variance

Describe Data Example 1

Describe Data Example 2

Describe Data Example 3

Describe Data Example 4

Describe Data Example 5

Which Variability Measures Should I Use?

• Is mean an appropriate central tendency measure for your distribution (unimodal, symmetric)?

  • Yes → Report the variability around the mean (variance or standard deviation)

    • Yes → use SD

    • Not necessarily, because I want to use Variance for more math → use Variance

  • No → Then, is median appropriate?

    • Yes → Report the variability around the median (Interquartile Range)

    • No → Settle for Range