Basic Statistical Concepts

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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

Distribution

1 2 3 4 5 6 7

1

2

N = 9

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

Distribution

What if we collected 200 observations instead of only 9?

…

Distribution

1 2 3 4 5 6 7

10

20

30

40

50

N = 200

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

Categorical Variables

VOTE

CuttingDoeSmithMissing

50

40

30

20

10

0

Cutting

Doe

Smith

Missing

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

Shape of a Distribution

Unimodal

Multimodal

Minor Mode

Major Mode

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

Computing a Mean

10 scores: 8, 4, 5, 2, 9, 13, 3, 7, 8, 5

ξΧ = 64

ξΧ/n = 6.4

Central Tendency

Is the mean always the best measure of central tendency?

No, skew pulls the mean in the direction of the skew

Central Tendency and Skew

Mode

Median

Mean

Central Tendency and Skew

Mode

Median

Mean

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)

Standard Deviation

m eanm ean m ean

low variability

small SD

high variability

large SD

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