• Agenda

• Descriptive Statistics

• Measures of Spread - Variability

Statistics

Descriptive Statistics

Statistics to summarize and describe the data we collected

Inferential Statistics

Statistics to make inferences from samples to the populations

Measures of Dispersion/ Spread/ Variability

• Indicates how cases vary or differ from each other

• Indicates how “central” or representative the center is

• Indicates how homogeneous or heterogeneous the variable is

Types of Measures of Dispersion

• Frequencies / Percentages

• Range / Interquartile Range

• Standard deviation / Variance

Gender Distribution: Group 1

Boy Girl(n=15) Gender

Num

ber o

f Kid

s

MODE = GIRL

Gender Distribution: Group 2

Boy Girl(n=15) Gender

Num

ber o

f Kid

s

MODE = GIRL

Frequency Distribution

• Frequency / Frequency count (f )• Percentages (%)• Proportions / Relative Frequency• Most common measures of dispersion

for nominal and ordinal variables• ONLY measures of dispersion for

nominal variables

Frequency Distribution

.9393%14Girl

.4747%7Boy

.5353%8Girl

.077%1Boy

ProportionPercentage

(%)Frequency

( f )

1

2

Gender Distribution: Group 2

Boy Girl(n=15) Gender

Num

ber o

f Kid

s

Gender Distribution: Group 3

Boy Girl(n=30) Gender

Num

ber o

f Kid

s

Frequency Distribution

.5353%8Girl

.4747%14Boy

.5353%16Girl

.4747%7Boy

ProportionPercentage

(%)Frequency

( f )

3

2

Range

• The distance between the highest score and the lowest score (highest – lowest)

• Can be used with numeric ordinal and interval/ratio variables

Example: Age

7 8 9 10 11

Group 1

7 8 9 10 11

Group 2 Median=10

Median=10

Example: Range

7 8 9 10 11

Group 1

7 8 9 10 11

Group 2

Range = 4 ( 7 to 11)

Measures of Dispersion / Spread

Nominal Ordinal

Frequency, %

Frequency, %

Frequency, %

Range, IQR Range, IQR

StandardDeviatn, Variance

Interval/Ratio

Deviation

• Difference from Standard

• In statistics,

difference from the mean

Distribution: Age

AGE(n=15)

7 8 9 10 11

Num

ber o

f Kid

s

Mean = 9.53- 2.53

-1.53 * 2

- 0.53 * 4 0.47 * 4

1.47 * 4

Age: Mean

- 2.53

- 1.53 * 2

- 0.53 * 4

1.47 * 4

0.47 * 4

Mean = 9.53

Sum of Negative Deviation = 7.7

Sum of Positive Deviation = 7.7

Variance / Standard Deviation

• Measures of dispersion for interval/ratio variables

• Variance (S2): Approximate average of the squared deviations from the mean

• Standard Deviation(S or SD): Square root of variance

• The larger the variance or SD is, the higher variability the data has

Variance

Xi = Value of Each case

X bar = Mean

N = Sample size

S 2 = )( xxi

N - 1

2

Standard Deviation

S =

Xi = Value of Each case

X bar = Mean

N = Sample size

1

)(

N

xXi 2

The Root-Mean-Square-Deviation or Standard Deviation

Root

Mean

Square

Deviation

n-1

( )

X - X

2

(X – X)2

n-1

Mean = X

n

Example: Age – Group 1

7 8 9 10 11AGE

Num

ber o

f Kid

s

(n=15)

Mean = 9

Example: Age – Group 2

7 8 9 10 11AGE

Num

ber o

f Kid

s

(n=15)

Mean = 9

Example: Age – Group 1

7 8 9 10 11AGE

Num

ber o

f Kid

s

(n=15)

Mean = 9

Example: Age – Group 2

7 8 9 10 11AGE

Num

ber o

f Kid

s

(n=15)

Mean = 9

Variance/Standard Deviation

Group 1

Group 2

What are the Mean and Standard deviation for the above distribution?

Why is Standard Deviation an important measure of spread?

Because it’s a sort of average of how much scores deviate around the mean.

High SDs indicate large variation in scores, or distributions that vary widely from the mean.

Because it has an important relationship with the normal distribution: THE 68-95-99.7 RULE.

-3s -2s -1s X +1s +2s +3s

Mean +/- 1 s d covers 68% of the population

Mean +/- 2 s d covers 95% of the population

Mean +/- 3 s d covers 99.7% of the population

The 68-95-99.7 Rule

68%

95%

99.7%

50%