CHAPTER 5 PP 110-130 Descriptive Statistics
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C H A P T E R 5
P P 1 1 0 - 1 3 0
Descriptive Statistics
Graphing data
Frequency distributions
Bar graphs Qualitative variable (categories)
Bars don’t touch
Histograms
Frequency polygons Quantitative variable (ordinal, interval, or ratio scale)
Others: Pie chart
Stem and leaf
Scatterplot
Example grade distribution
A 1
A- 2
B+ 3
B 7
B- 8
C+ 6
C 3
C- 2
D 1
F 1
N = 34
Class interval
frequency distribution
Graphs!
Read X and Y axis carefully
Death Rates in America
40
60
80
100
120
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
Year
Nu
mb
er p
er 1
00,0
00 p
op
ula
tion
Age 1-4
Age 15-24
Example bar graph Rauscher, Shaw, & Ky (1993). Mozart Effect
119
111 110
N = 36 college students
Lots of cool graphs!
Florence Nightingale’s “coxcomb” diagram Blue: died of sickness; Red: died of wounds; Black: died of other causes
Graph interpretation
Recency
effect Reminiscence
bump
Careful to read values on each axis – graphs can be deceiving!
Descriptive statistics
Data collected in a study = raw data
Reports of a study = summary data Descriptive statistics provide that summary
Measures of central tendency Describe “middleness” of distribution of scores
Mean
Median
Mode
Measures of variation Describe width or dispersion of a distribution
Range
Standard deviation
Variance
Descriptive statistics
Measure of central tendency
Mean
Mean for population = sum of scores ÷ # of scores in distribution
𝜇 =Σ𝑋 𝑁
Mean for sample = sum of scores ÷ # scores in distribution
M or X =Σ𝑋 𝑁
Mean as the balance point
The mean balances the distances (or deviations) of all scores
Mean
5
5
5
5
Scores (x)
2
2
6
10
Distance from mean
-3
-3
1
5
______ ∑X = 20 N = 4 M = 5
_____ ∑X = 0
Effect of changing 1 score
The mean is not a “robust” statistic It is highly influenced by a single outlier score
26
29
31
32
34
35
38
40
42
83
390
390 / 10
39
____ ∑X = ∑X / N = M =
26
29
31
32
34
35
38
40
42
33
340
340 / 10
34
____ ∑X = ∑X / N = M =
Adding a constant
If you add, subtract, multiply or divide all scores by constant:
The same change is made to M
26
29
31
32
34
35
38
40
42
83
390
390 / 10
39
36
39
41
42
44
45
48
50
52
93
490
490 / 10
49
X + 10
Descriptive statistics
Measure of central tendency
Mean
Mean for population = sum of scores ÷ # of scores in distribution
µ = ∑ X / N
Mean for sample = sum of scores ÷ # scores in distribution
X or M = ∑ X / N
Median
Middle score in distribution
Order scores from highest to lowest
If N is even number, average the two middle scores
Calculating the median for RTs
512 587 590 578 567 533 573 529 577 572 572 591 575 577 534
512 529 533 534 567 572 572 573 575 577 577 578 587 590 591
512 587 590 578 567 533 573 529 899 572 572 591 575 577 534
512 587 590 578 567 533 573 529 177 572 572 591 575 577 534
sorted scores
573 564.47
Median
Mean
573 585.93
572 537.80
Add Hi X Add Lo X
Median is
a robust
statistic!
Descriptive statistics
Measure of central tendency
Mean
Mean for population = sum of scores / # of scores in distribution
µ = ∑ X / N
Mean for sample = sum of scores / # scores in distribution
X or M = ∑ X / N
Median
Middle score in distribution
Order scores from highest to lowest
If N is even number, average the two middle scores
Mode
Score that occurs with greatest frequency
Example grade distribution
A 1
A- 2
B+ 3
B 7
B- 8
C+ 6
C 3
C- 2
D 1
F 1
N = 34
M = 80.38
Median = 81
Mode = B-
Can have 2+ modes
Sample grade distribution with 2 modes
0
1
2
3
4
5
6
7
A A- B+ B B- C+ C C- D F
Types of distributions
Normal distribution
Bell-shaped
Symmetrical
Only 1 mode
Mean, median, mode all equal
Kurtosis: spread of distribution
How flat or peaked
Mesokurtic: medium peak (like normal distribution)
Leptokurtic: tall and thin
Platykurtic: flat and broad
Measures of central tendency Indicators of the shape of the distribution
How mean, median, and mode change w/ shape of distribution
Normal distribution
Positive skew
Tail to positive scores
Negative skew
Tail to negative scores Positive skew Negative skew
Which measure of central tendency to use?
If interval or ratio data and normally-distributed
Use mean
If interval or ratio data and there are outliers or a skewed-distribution
Use median
If nominal data
Use mode But, that’s not enough info …
Measures of variation
Range
Difference between lowest and highest scores in a distribution
= Maximum score – minimum score
Easily distorted by an outlier (low or high score)
Standard deviation
Average distance of scores in a distribution from the mean
If sum deviations from mean = zero! SO…
Average deviation: Use absolute values
Standard deviation: Use squared deviation scores
For population: 𝜎 =Σ(𝑋−𝜇)2
𝑁
Example grade distribution
A 1
A- 2
B+ 3
B 7
B- 8
C+ 6
C 3
C- 2
D 1
F 1
N = 34
M = 80.38
Median = 81
Mode = B-
s = 7.92
M = 80.38 M + s = 88.3 M - s = 72.5
Note: most scores are w/in 8 pts of mean
Calculating standard deviation (σ)
RTs x - M (x - M)2
512 -52.47 2753.101
587 22.53 507.6009
590 25.53 651.7809
578 13.53 183.0609
567 2.53 6.4009
533 -31.47 990.3609
573 8.53 72.7609
529 -35.47 1258.121
577 12.53 157.0009
572 7.53 56.7009
572 7.53 56.7009
591 26.53 703.8409
575 10.53 110.8809
577 12.53 157.0009
534 -30.47 928.4209
Avg =
564.4667 8593.734 sum of (X-M)2
572.9156 Variance: sum divided by N
23.93565 SD: square root of sum/N
1. Calculate deviation score (score – mean)
2. Square deviations
3. Sum squared deviations
4. Divide by N 1. N = # of scores
2. This step = variance
5. Take square root of value
Calculating standard deviation (s)
1. Calculate deviation score (score – mean)
2. Square deviations
3. Sum squared deviations
4. Divide by N or N - 1 1. This step = variance
2. Use N for population
3. Use N-1 to estimate population from sample
5. Take square root of value
RTs x - M (x - M)2
512 -52.47 2753.101
587 22.53 507.6009
590 25.53 651.7809
578 13.53 183.0609
567 2.53 6.4009
533 -31.47 990.3609
573 8.53 72.7609
529 -35.47 1258.121
577 12.53 157.0009
572 7.53 56.7009
572 7.53 56.7009
591 26.53 703.8409
575 10.53 110.8809
577 12.53 157.0009
534 -30.47 928.4209
Avg =
564.4667 8593.734 sum of (X-M)2
sd = 613.8381 Variance: sum divided by N-1
24.77576 = 24.77576 SD: square root of sum/N-1
Measures of variation
Standard deviation of population
𝜎 =Σ(𝑋−𝜇)2
𝑁
Standard deviation of sample (when estimating population)
s =Σ(𝑋−𝑀)2
𝑁−1
Variance
Population = σ2 = Σ(𝑋−𝜇)2
𝑁 or sample = s2 =
Σ(𝑋−𝜇)2
𝑁
Why use N – 1 ?
Sample is less variable than the population Divide by smaller # so yields more conservative estimate
of variance or SD Makes variance score larger
Use n-1 so can make conclusions about population (not just describe your sample)
Thank you, Excel!
For example, if data is in column B from row 1 to 20
Sum: =sum(B1:B20)
Mean: =average(B1:B20)
Median: =median(B1:B20)
Mode: =mode(B1:B20)
Maximum score: =max(B1:B20)
Minimum score: =min(B1:B20)
Range: Subtract Max score from Min score
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