Distributions

Discrete Variable

Frequency tables

• containing the number of occurrences in each class of data
• often used to create histograms and frequency polygons

Colour	Frequency
Brown	17
Yellow	18
Red	7
Green	7
Blue	2
Orange	4

Frequency Distribution

• the distribution of empirical data
• consists of a count of the number of occurrences of each value
• For a discrete random variable, a probability distribution contains the probability of each possible outcome
• The sum of all probabilities is always 1.0

Frequency Distribution	Probability Distribution

Continuous Variable Distribution

Problems?

Response time (in millisecond)
568 577 581 640 641 645 657
673 696 703 720 728 729 777
808 824 825 865 875 1007

Grouped Frequency Distribution

• a frequency distribution in which frequencies are displayed for ranges of data rather than for individual values.
• Histogram is a graphical representation of a distribution .
• It partitions the variable on the x-axis into various contiguous class intervals of (usually) equal widths.

Example

Range	Frequency
500-600	3
600-700	6
700-800	5
800-900	5
900-1000	0
1000-1100	1

A probability density function

A probability density function is a formula that can be used to compute probabilities of a range of outcomes for a continuous random variable.

Normal Distribution

• one of the most common continuous distributions
• sometimes referred to as a "bell-shaped distribution.

Skewness

A distribution is skewed if one tail extends out further than the other.

• A distribution has positive skew (is skewed to the right) if the tail to the right is longer
• A distribution has a negative skew (is skewed to the left) if the tail to the left is longer

Kurtosis

• Leptokurtic is a distribution with long tails relative to a normal distribution
• Platykurtic is a distribution with short tails relative to a normal distribution

Quiz

1 A frequency distribution contains the frequency of every value in the distribution.

	True
	False

Answer >>

True

The distribution of empirical data is called a frequency distribution and consists of a count of the number of occurrences of each value.

2 A grouped frequency distribution should be used instead of a frequency distribution when the

	distribution is bimodal.
	distribution is skewed.
	variable is continuous.

Answer >>

variable is continuous.

When a variable is truly continuous, each value will have a frequency of 1. Therefore, grouped frequency distributions are needed with continuous variables.

3 A symmetric distribution

	has equal positive and negative skews.
	has no skew.
	can have either positive or negative skew, but not both.

Answer >>

has no skew.

In a symmetric distribution, the tails extend equally in both directions. Therefore, there is no skew.

4 The following distribution has

	a positive skew.
	a negative skew.
	no skew.

Answer >>

a positive skew.

The tail in the positive direction is longer than the tail in the negative direction, thus it has a positive skew.

5 The area under the curve of a probability distribution is

Answer >>

1

The area is 1 by definition, meaning that the probability that a score chosen at random will occur under the curve is 1.

6 A normal or bell-shaped distribution has its greatest probability density in its tails.

	True
	False

Answer >>

False

The distribution is higher and therefore denser in the middle of the distribution.

7 Which of the following distributions is/are symmetric?

	A
	B
	C
	D

Answer >>

A, D

A and D are symmetric, meaning if you folded them in the middle, the two sides would match perfectly. Distributions B and C have positive skew.