Transcript class 13 stats review

Statistics Review I
Class 14
WHAT WOULD YOU LIKE TO KNOW?
STATISTICS AS VOX POPULI,
THE VOICE OF THE PEOPLE
STATISTICAL SKILLS AND DISCOVERY
CLASS OVERVIEW
Levels of Measurement
Measures of Centrality and Dispersion
* Centrality (mean, median, mode)
* Dispersion (range, variance, std. deviation, std. error)
* Z scores and Z distribution
Confidence Intervals
Exploring Data Sets
* Reasons
* Methods (histograms, features of distributions)
Dealing with Outliers
LEVELS OF MEASUREMENT
1. Categorical
2. Ordinal
3. Continuous
a. Interval
b. Ratio
c. Discrete
Categorical Variables
1. Refer to categories: human, cat, eggplant
2. All or none: Can’t be 1 third human, 2 thirds eggplant
3. Numbers serve as labels, not values: 1 = human, 2 = eggplant
“1” is not less than “2”; human is not less than eggplant
4. Common kinds of categorical variables: gender, race, major
5. Binary: only two values: Yes/No, Day/Night, present/absent
6. Non-Binary: Multiple values. Animal, vegetable, mineral
Democrat, Republican, Independ.
7. Nominal: Values are known signifiers:
“Did Joey go potty? Yes? Was it Number 1 or Number 2?”
In some sports, numbers on jerseys represent player
position; e.g. 1 = tackler, 2 = runner, etc.
Ordinal Variables
Numeric values refer to the ordering of things
Rankings: 1 = First place, 2 = second place
Chronology: 1= occurred first, 2 = occurred second, etc.
Numeric valued DO NOT indicate how much “1” differs from “2”
Bike race: 1st place (27.24); 2nd place (27.28); 3rd place (33.10)
Grant scores:
1. 99.89
2. 92.63
winners
3. 89.76
4. 89.75
5. 88.84
6. 79.48
losers
CONTINUOUS VARIABLES
Interval:
Discrete:
Most stat tests rely on interval data
Equal intervals represent equal differences
Virtually same as "interval" but there is a finite range of
values, as in Likert scales.
“How happy are you with your cell phone service?”
1
2
3
4
5
Not at all Barely somewhat Very
Greatly
Ratio:
Ratios of values on scale are meaningful
Must have meaningful “0” point
Likert scale above NOT ratio, b/c 2:4 ≠ 1:2
Temperature, RT, number of yawns in class ARE ratio
GUESS THAT VARIABLE
Example
Variable
1 = female, 2 = male
Categorical, binary
32.75 miles per gallon
Ratio
1 = slightly tired 2 = moder. tired 3 = very tired
Interval
352 Smith Hall
Categorical, non-binary
Top 4 Reasons to Learn Stats:
Ordinal
1.
2.
3.
4.
Necessary for career
Source of serenity
Great ice-breaker
Fun for whole family
Distress and Disclosure:
A Sample Experiment That Never Occurred!!!
Hyp: Increased anxiety leads
to disclosure.
Ss see scary movie or neutral
movie.
Ss asked to rate how scary
they found the movie.
Ss write about thoughts and
feelings movie created.
Measures of Centrality
MODE
Most frequent value, occurrence
MEDIAN
Middle-most value; 50% above/below
MEAN
Arithmetic average
How many words written after seeing scary movie?
Number of words written: 2, 2, 3, 5, 8
MODE = ?
2
MEDIAN = ?
3
MEAN = ?
4
[2 + 2 + 3 + 5 + 8 / 5 = 4]
Relations Btwn Mean, Median, Mode
Number of words written?
N=5
5
Mode
Median
Mean
4
3
2
N = 5: 1, 2, 2, 3, 8
2.0
3.0
3.8
1
0
$1
$2
$3
$4
$5
$6
$7
$8
$5
$6
$7
$8
$5
$6
$7
$8
N = 10
5
N = 10: 1, 2, 3, 3, 3, 4, 5,
5, 6, 8
4
3.0
3.5
4.0
3
2
1
0
$1
$2
$3
$4
N = 20
N = 20: 1, 2, 2, 3, 3, 3, 4, 4,
4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 7, 8
4.0
4.0
4.35
5
4
3
2
1
0
$1
How does change in N affect rel.
btwn Mean, Median, and Mode?
$2
$3
$4
If true distribution is normal, then as sample
increases mean, median, and mode converge.
MEASURES OF DISPERSON
Mode
N = 20: 1, 2, 2, 3, 3, 3, 4, 4,
4, 4, 4, 5, 5, 5,5, 6, 6, 6, 7, 8
Median
4.0
4.0
Range: Difference between highest score
and lowest score.
Deviation (from mean), AKA “Error”:
Difference between individual score and mean
Sum of Squared Errors (SS): Why? To get
a meaningful index of average dispersion.
Mean
4.35
8 – 1 = 7 = range
8 – 4.35 = 3.65 = 8’s deviation
1 - 4.35 + 2 – 4.35 ... + 7 – 4.35 +
8 – 4.35 = 0.
Useless!
(1 - 4.35)2 + (2 – 4.35)2 ...+ (7 – 4.35)2
+ (8 – 4.35)2 = 87.00.
Useful!
Variance and Standard Deviation
We need to get an estimate of average dispersion from mean, just like
the mean gives an estimate of average score.
Variance = s2 = Average deviation in sample = SS
N-1
Two problems with variance:
87 = 4.58 = s2
20-1
1) units, based on sq’d deviations, are not relatable to actual scores.
2) Variance tends to be a large, unwieldy, number.
Standard Deviation = s =
s2
= sq. root of variance =
1 sd above and below mean = 68% of distribution
2 sd above and below mean = 95% of distribution
4.58 = 2.14
Z Scores and Z Distribution
Mean
SD
DV 1: “How anxious were you during movie?”
4.23
2.71
DV 2: Number of words written about movie.
28.71
11.65
discrete data
ratio data
Issue: How do we compare anxiety with word production?
Z-score conversion: Effect is to convert different metrics into a common metric
Z=X–X
s
Sub. 24: anxious = 3; words = 22
Z_anxious = 3 – 4.23 = -.45
2.71
Z distribution is normal, mean = 0, SD = 1
SPSS: Descriptives,
“Save standardized values as variables”
Z_words = 22 – 28.71 = -.58
11.65
Standard Error of the Mean
Sample mean ( X ) estimates true population mean (µ)
Many sample means from same population will vary.
Standard Error of the Mean (SE) = the average amount that
sample means vary around true mean.
If n of sample mean ≥ 30, SE can be estimated based on s (std.
deviation), and sample n.
Formula for SE:
SE X = s/√n
SE Movie anxiety study: DV = reported anxiety; n = 43, s = 2.71
SE = (2.71 / √43) = 0.41
Note: SE is much smaller than SD. Why?
CONFIDENCE INTERVALS
Issue: How do we know if the sample mean is a good estimate of the true
mean? In other words, how do we estimate a mean’s accuracy?
Confidence Intervals (CI) estimate accuracy of sample means.
CI shows boundary values (highest & lowest) w/n which true mean
is likely to occur.
Conventional boundary captures true mean 95% of time.
Calculation: Upper boundary =
Lower boundary =
X + (1.96 * SE)
X − (1.96 * SE)
Movie anxiety study: X = 4.23, SE = 0.41
Lower CI = 4.23 - (1.96 * 0.41) = 3.43
Upper CI = 4.23 + (1.96*.041) = 5.03
GRAPHIC
REPRESENTATION OF CI
6
Anxiety Rating
5
4
Alone
With Friend
3
2
1
0
Neutral Movie
Scary Movie
Error bars overlap; means are likely
from same distribution.
Error bars DON’T overlap; means
are likely from different distributions
Differences are not meaninful.
Differences are meaningful
GRAPHICALLY EXPLORING DATA USING
CENTRALITY AND DISPERSION
Why explore data?
1. Get a general sense or feel for your data.
2. Determine if distribution is normal, skewed, kurtotic, or
multi-modal (more on this soon).
3. Identify outliers
4. Identify possible data entry errors
DATA BUGS ARE A HAZZARD:
KNOW WHAT'S IN YOUR DATA!
=
+
12, 19, 17, 14,
17, 13, 17, 15
+
147
=
Normally Distributed Data Set
SPSS output: Note similarity
between mean, median, mode
Skewed Distribution
Positive Skew
Possible
"floor effect"
Negative Skew
Possible
"ceiling effect"
Kurtosis
Neuroticism Measure
Positive kurtosis,
“leptokurotic”
Problems?
"Normativity bias?"
DV doesn't discriminate
IV wasn't impactful
Drinks Per Week
Negative kurtosis,
“platykurotic”
Problems?
Distinctiveness bias?
IV and/or DV too ambiguous
Population too diverse
Bimodality
Note: What clues in “statistics” output that the distribution may be bimodal?
Bimodality suggests 2 (or more) populations
Multimodal: More than two modes.
Outliers
BOX AND WHISKER GRAPH
Top 25%
Upper Quartile
Median (50 %)
Lower Quartile
Bottom 25%
BOX AND WHISKER GRAPH, AND DATA CHECKING
subject number
Detecting
Skew
Detecting
Outliers
DEALING WITH OUTLIERS
1. Check raw data: Entry problem? Coding problem?
2. Remove the outlier:
a. Must be at least 2.5 DV from the mean (some say 3 DV)
b. Must declare deletions in pubs.
c. Try to identify reason for outlier (e.g., other anomalous responses).
3. Transform data: Convert data to a metric that reduces deviation. (More on this
in next slide).
4. Change the score to a more conservative one (Field, 2009):
a. Next highest plus 1
b. 2 SD or 3 SD above (or below) the mean.
c. ISN’T THIS CHEATING? No (says Field) b/c retaining score biases outcome.
Again, report this step in pubs.
5. Run more subjects!
Data Transformations
1. Log Transformation (log(X)): Converting scores into Log X reduces
positive skew, draws in scores on the far right side of distribution.
NOTE: This only works on sets where lowest value is greater than 0.
Easy fix: add a constant to all values.
2. Square Root Transformation (√X): Sq. roots reduce large numbers
more than small ones, so will pull in extreme outliers.
3. Reciprocal Transformation (1/X): Divide 1 by each score reduces
large values. BUT, remember that this effectively reverses valence, so
that scores above the mean flip over to below the mean, and vice versa.
Fix: First, preliminary transform by changing each score to highest
score minus the target score. Do it all at same time by 1/(Xhighest – X).
4. Correcting negative skew: All steps work on neg. skew, but first must
reverse scores. Subtract each score from highest score. Then, rereverse back to original scale after transform completed.