Statistics for Everyone Workshop Fall 2010

Part 2

Descriptive Statistics:

Measures of Central Tendency and Variability

Workshop presented by Linda Henkel and Laura McSweeney of Fairfield University

Funded by the Core Integration Initiative and the Center for Academic Excellence at Fairfield University

Descriptive Statistics

Once you know what type of measurement scale the data were measured on, you can choose the most appropriate statistics to summarize them:

Measures of central tendency: Most representative score

Measures of dispersion: How far spread out scores are

Measures of Central Tendency

Central tendency = Typical or representative value of a group of scores

• Mean: Average score• Median: middlemost score; score at 50th

percentile; half the scores are above, half are below

• Mode: Most frequently occurring score(s)

Measures of Central Tendency

Measure Definition

Takes Every Value Into Account? When to Use

Mean M = X / n Yes Numerical data

BUT… Can be heavily influenced by outliers so can give inaccurate view if distribution is not (approximately) symmetric

Median Middle value No Ordinal data or for numerical data that are skewed

Mode Most frequent data value

No Nominal data

Three Different Distributions That Have the Same Mean

Sample A Sample B Sample C

0 8 6

2 7 6

6 6 6

10 5 6

12 4 6

Mean 6 6 6

Measures of Variability

Knowing what the center of a set of scores is is useful but….

How far spread out are all the scores?

Were all scores the same or did they have some variability?

Range, Standard deviation, Interquartile range

Variability = extent to which scores in a distribution differ from each other; are spread

out

The Range as a Measure of Variability

Difference between lowest score in the set and highest score

• Ages ranged from 27 to 56 years of age

• There was a 29-year age range

• The number of calories ranged from 256 to 781

Sample Standard Deviation

1N

MX2

Standard deviation = How far on “average” do the scores deviate around the mean?

s = SD =

• In a normal distribution, 68% of the scores fall within 1 standard deviation of the mean (M SD)

• The bigger the SD is, the more spread out the scores are around the mean

Variations of the Normal Curve (larger SD = wider spread)

Interquartile Range

Quartile 1: 25th percentileQuartile 2: 50th percentile (median)Quartile 3: 75th percentileQuartile 4: 100th percentile

Interquartile range = IQR =Score at 75th percentile – Score at 25th percentile

So this is the midmost 50% of the scores

Interquartile Range on Positively (Right) Skewed Distribution

IQR is often used for interval or ratio data that are skewed

(do not want to consider ALL the scores)

Measures of Variability

Measure Definition

Takes Every Value Into Account? When to Use

Range Highest -lowest score

No, only based on two most

extreme values

To give crude measure of spread

Standard Deviation

68% of the data fall

within 1 SD of the mean

Yes, but describes majority

For numerical data that are approximately

symmetric or normal

Interquartile Range

Middle 50% of the data fall within the IQR

No, but describes

most

When numerical data are skewed

Presenting Measures of Central Tendency and Variability in Text

The number of fruit flies observed each day ranged from 0 to 57 (M = 25.32, SD = 5.08).

Plants exposed to moderate amounts of sunlight were taller (M = 6.75 cm, SD = 1.32) than plants exposed to minimal sunlight (M = 3.45 cm, SD = 0.95).

The response time to a patient’s call ranged from 0 to 8 minutes (M = 2.1, SD = .8)

Sentences should always be grammatical and sensible. Do not just list a bunch of numbers. Use the statistical information to supplement what you are saying

Presenting Measures of Central Tendency and Variability in Tables

(Symmetric Data)

Range M SD

Number of Fruit Flies 0 to 57 25.32 5.08

Weight (lbs) 118 to 208 160.31 10.97

Response time to Patient’s Call (mins)

0 to 8 2.1 .8

Be sure to include the units of measurement! You can include an additional column to put the sample size (N)

Presenting Measures of Central Tendency and Variability in Tables

(Skewed Data)

Range Median IQR

Number of Fruit Flies 0 to 57 27 9

Weight (lbs) 118 to 208 155.6 12

Response time to Patient’s Call (mins)

0 to 8 1.5 1

Be sure to include the units of measurement! You can include an additional column to put the sample size (N)

What’s the Difference Between SD and SE?

Sometimes instead of standard deviation, people report the standard error of the mean (SE or SEM) in text, tables, and figures

Standard deviation (SD) = “Average” deviation of individual scores around mean of scores

• Used to describe the spread of your (one) sample

Standard error (SE = SD/N) = How much on average sample means would vary if you sampled more than once from the same population (we do not expect the particular mean we got to be an exact reflection of the population mean)

• Used to describe the spread of all possible sample means and used to make inferences about the population mean

Standard error usually looks better in figures because it is not as large

• Dangers of low N: Be sure to emphasize to students that with a small sample size, data may not be representative of the population at large and they should take care in drawing conclusions

• Dangers of Outliers: Be sure your students look for outliers (extreme values) in their data and discuss appropriate strategies for dealing with them (e.g., eliminating data because the researcher assumes it is a mistake instead of part of the natural variability in the population = subjective science)

Teaching Tips

Finding descriptive statistics

Teaching tips:

• Hands-on practice is important for your students

• Sometimes working with a partner helps

Time to Practice