Review

Which of these is a parameter?

A. The average height of all people

B. The time it takes rat #3 to learn the maze

C. The number of subjects in your experiment

D. The average memory score of subjects in your sample

Review

You want to know the average time people can hold their breath. You measure 10 people and find that their average is 104 seconds, which you treat as a guess for people in general. This is a(n)

A. Descriptive statistic

B. Estimator

C. Inferential statistic

D. Parameter

Review

Curious how heavy your dishes are, you weigh each one and then calculate the average. This is a(n)

A. Descriptive statistic

B. Estimator

C. Inferential statistic

D. Parameter

Distributions

9/4

Outline

• Distributions• Frequency• Histograms• Cumulative frequency• Quantiles• Continuous variables• Shape of a distribution

Distribution

• The set of values present in a sample or population– Which values occur– How often

• Starting point for statistics– Every statistic is computed from sample distribution– Every parameter is a property of population distribution

• Need ways of representing or talking about distributions

Frequency

• Easiest way to characterize distribution• How often each value occurs

f(x) = frequency of value xSample: {1, 6, 3, 8, 6, 4}. f(6) = ?

• Frequency table– Shows frequencies of all values– 1st column for value,

2nd column for frequencyx f(x)

2

3

1

4

5 6

11 1

7 4{5,7,3,7,2,5,5,3,7,5,3,11,7,5,3,5}

Histogram

Units

Variable Label Values

Frequencyof this value

Graphical representation of a distribution, showing frequency of each value

Cumulative Frequency

• Number of scores below or equal to a given valueF(x) = cumulative frequency for value x

{4,3,4,5,3,4,2,4,3,4}

f(3) = ?F(3) = ?

34

x f(x) F(x)

2

3

1

3

4 5

5 1

1

4

9

10f(3)

f(4)

Quantile

• Quantile - the value of X that's greater than a certain fraction of the data

• Percentile - quantile defined by a certain percentage

90th %ile25th %ileInterpolation

{1,2,2,4,5,5,7,8,8,8}

{8,2,5,5,7,1,8,2,4,8}

50th percentile = 590th percentile = 8

Continuous vs. Discrete Variables

• Discrete variable– Can only take certain values (usually integers)– Counts: people, test score, stories, …

• Continuous variable– Infinite set of values, in principle– Height, weight, temp, IQ, …– For any two scores, there are other possible

scores in between

0123456789

10

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74

Height (Inches)

Fre

qu

ency

(st

ud

ents

)

71.5

72.5

73.5

Histograms of Continuous Variables

• Plotting unique scores isn’t useful• Bins or intervals

– Ranges for grouping continuous variables– Best width depends on number of data

Density

100%

2%

• Frequency only well-defined for discrete variables– f(x): scores exactly equal to x– 0 almost everywhere for continuous variables

• Density function– Describes theoretical distribution of continuous variable– Allows determination of number of scores in any range, by integration– Usually shown as proportion of total population (probability), not frequency

Household Income

Den

sity

Shape of a Distribution• Information beyond average score & variability

– Broad, often qualitative property

• Need "nice" shape to do statistics• Normal distribution

– Gold standard for good shape– Symmetric, unimodal, thin tails

Bad Shape• Skew: Asymmetric distribution

– Extreme scores in one direction bias results

• Positive skew vs negative skew - which tail is bigger• Solutions

– Only consider order of scores (“ordinal data”)– Transform: Do statistics on new variable

Bad Shape• Multimodal: More than one peak• Suggests there are multiple constituent populations

– Learners vs. non-learners

• Solution: discretize– Do statistics on proportion of learners

Review

Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9}

What is f(4)?

A. 2

B. 4

C. 6

D. 8

Review

Data: {2, 5, 6, 8, 5, 6, 4, 3, 2, 1, 4, 9}

What is the 75th percentile?

A. 2

B. 6

C. 8

D. 9

Review

Find the bimodal distribution

A.C.

B.D.

Score

Den

sity

Score

Den

sity

Score

Den

sity

Score

Den

sity