basic statistical concepts . so, you have collected your data … now what? we use statistical...
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So, you have collected your data …
Now what?We use statistical analysis to test our hypotheses make claims about the population
This type of analyses are called inferential statistics
But, first we must …
Organize, simplify, and describe our body of data (distribution).
These statistical techniques are called descriptive statistics
Distributions
Recall a variable is a characteristic that can take different values
A distribution of a variable is a summary of all the different values of a variable Both type (each value) and token (each
instance)
Distribution
How excited are you about learning statistical concepts?
1 2 3 4 5 6 7
Comatose Hyperventilating
1 2 2 3 4 4 5 6 7
7 Types: 1,2,3,4,5,6,7
9 Tokens: 1,2,2,3,4,4,5,6,7
Properties of a Distribution Shape symmetric vs. skewed unimodal vs. multimodal
Central Tendency where most of the data are mean, median, and mode
Variability (spread) how similar the scores are range, variance, and standard
deviation
Representing a Distribution
Often it is helpful to visually represent distributions in various ways Graphs continuous variables (histogram, line graph) categorical variables (pie chart, bar chart)
Tables frequency distribution table
Continuous Variables
13
8
11
1817
1210
75
02468
101214161820
50-54
55-59
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-100
Exam scores
Fre
qu
ency
Frequency Distribution Table
VAR00003
2 7.7 7.7 7.7
3 11.5 11.5 19.23 11.5 11.5 30.85 19.2 19.2 50.0
4 15.4 15.4 65.42 7.7 7.7 73.14 15.4 15.4 88.52 7.7 7.7 96.2
1 3.8 3.8 100.026 100.0 100.0
1.00
2.003.004.00
5.006.007.008.00
9.00Total
ValidFrequency Percent Valid Percent
CumulativePercent
Shape of a Distribution
Symmetrical (normal) scores are evenly distributed about
the central tendency (i.e., mean)
Shape of a Distribution
Skewed extreme high or low scores can skew
the distribution in either direction
Negative skew
Positive skew
Distribution
So, ordering our data and understanding the shape of the distribution organizes our data Now, we must simplify and describe the distribution What value best represents our distribution? (central tendency)
Central Tendency
Mode: the most frequent score good for nominal scales (eye color) a must for multimodal distributions
Median: the middle score separates the bottom 50% and the
top 50% of the distribution good for skewed distributions (net
worth)
Central Tendency
Mean: the arithmetic average add all of the scores and divide by total
number of scores This the preferred measure of central
tendency (takes all of the scores into account)
X
N
X X
npopulation
sample
Central Tendency
Is the mean always the best measure of central tendency?
No, skew pulls the mean in the direction of the skew
Distribution
So, central tendency simplifies and describes our distribution by providing a representative score
What about the difference between the individual scores and the mean?(variability)
Variability
Range: maximum value – minimum value only takes two scores from the distribution into
account easily influenced by extreme high or low scores
Standard Deviation/Variance the average deviation of scores from the mean
of the distribution takes all scores into account less influenced by extreme values
Standard Deviation
most popular and important measure of variability a measure of how far all of the individual scores in the distribution are from a standard (mean)
Computing a Standard Deviation
10 scores: 8, 4, 5, 2, 9, 13, 3, 7, 8, 5
ξΧ/n = 6.4
8 – 6.4 =
4 – 6.4 =
5 – 6.4 =
2 – 6.4 =
9 – 6.4 =
13 – 6.4 =
3 – 6.4 =
7 – 6.4 =
8 – 6.4 =
5 – 6.4 =
1.6
- 2.4
- 1.4
- 4.4
2.6
6.6
- 3.4
0.6
1.6
- 1.4
2.56
5.76
1.96
19.36
6.76
43.56
11.56
0.36
2.56
1.96
SS = 96.4
variance = 2 = SS/N
10.71
2 X 2N
standard deviation = =
2 X 2N
standard deviation = =
3.27
Standard Deviation
In a perfectly symmetrical (i.e. normal) distribution 2/3 of the scores will fall within +/- 1 standard deviation
6.4
+1
-1
9.673.13
Variance vs. SD So, SD simplifies and describes the distribution by providing a measure of the variability of scores If we only ever report SD, then why would variance be considered a separate measure of variability?Variance will be an important value in many calculations in inferential statistics
Review Descriptive statistics organize, simplify, and describe the important aspects of a distribution This is the first step toward testing hypotheses with inferential statistics Distributions can be described in terms of shape, central tendency, and variabilityThere are small differences in computation for populations vs. samplesIt is often useful to graphically represent a distribution
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