Statistics for Everyone Workshop Fall 2010
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Transcript of Statistics for Everyone Workshop Fall 2010
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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
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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
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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)
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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
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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
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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
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Variability = extent to which scores in a distribution differ from each other; are spread
out
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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
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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
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Variations of the Normal Curve (larger SD = wider spread)
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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
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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)
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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
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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
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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)
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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)
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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
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• 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
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Finding descriptive statistics
Teaching tips:
• Hands-on practice is important for your students
• Sometimes working with a partner helps
Time to Practice