DAY 3

14 Jan 2014

Today is

A. January 14, 2014

B. January 13, 2013

Recap

• Organizing Data• Qualitative & Quantitative Data• Frequency distribution & relative frequency

distribution• Single-value grouping, Limit grouping, Cut point

grouping• Histogram, Dotplots, Stem-and-leaf diagrams

Objective of the day:

• Distribution shapes

• Descriptive Measures => Central Measures => Mean, Median, Mode.

• Measures of Variations => Standard Deviation

Section 2.4

Distribution Shapes

Definition 2.10

Distribution of a Data Set

The distribution of a data set is a table, graph, or formula that provides the values of the observations andhow often they occur.

Relative-frequency histogram and approximating smooth curve for the distribution of heights

Common distribution shapes

Relative-frequency histogram for household size

Identify the shape of the distribution.

Example:

Relative-frequency histogram for household size

Identify the shape of the distribution.

Example:

Definition 2.12

Population and Sample Distributions; Distribution of a Variable

The distribution of population data is called the population distribution, or the distribution of the variable.

The distribution of sample data is called a sample distribution.

Population distribution and six sample distributions for household size

Key Facts: Population and Sample Distributions

For a simple random sample, the sample distributionapproximates the population distribution (i.e., thedistribution of the variable under consideration). The larger the sample size, the better the approximationtends to be.

Chapter 3

Descriptive Measures

Number that describe data set.

Descriptive Measures

Section 3.1

Measures of Center

Measure of Center

Descriptive measures that indicates where the center or most typicalvalue of data set lies are called measure of central tendency or measures of center.

Three most important measures of center:1. Mean2. Median3. Mode

Definition 3.1

Mean of a Data Set

The mean of a data set is the sum of the observations divided by the number of observations.

mean = sum of the observations / the number of observations.

Data Set I Data Set II

Example:

Data Set I Data Set II

Means in Data Set I and Data Set II

Example:

Definition 3.2

Median of a Data Set

Arrange the data in increasing order.

• If the number of observations is odd, then the median is the observation exactly in the middle of the ordered list.

• If the number of observations is even, then the median is the mean of the two middle observations in the ordered list.

In both cases, if we let n denote the number of observations, then the median is at position (n + 1) / 2 in the ordered list.

Definition 3.3

Mode of a Data Set

Find the frequency of each value in the data set.

• If no value occurs more than once, then the data set has no mode.

• Otherwise, any value that occurs with the greatest frequency is a mode of the data set.

Data Set I

Median in Data Set I

Example:

Data Set I

300 300 300 300 300 300 400 400 450 450 800 940 1050

Median is at the position (n+1)/2 = (13+1)/2 = 7

Median = ?

Data Set I

Median in Data Set I

Example:

Data Set I

300 300 300 300 300 300 400 400 450 450 800 940 1050

Median is at the position (n+1)/2 = (13+1)/2 = 7

Median = 400

Data Set I

Mode in Data Set I

Example:

Data Set I

300 300 300 300 300 300 400 400 450 450 800 940 1050

Mode = ?

Data Set I

Mode in Data Set I

Example:

Data Set I

300 300 300 300 300 300 400 400 450 450 800 940 1050

Mode = 300

Data Set I Data Set II

Mean, Median, and Mode in Data Set I and Data Set II

Example:

Definition 3.4

COMPARISON OF MEAN, MEDIAN, MODE: �

1. Note that the mean is pulled in the direction of the skewness, i.e. in the direction of the extreme observation. The mean is sensitive to extreme observations (very large or very small in comparison to the rest of the data). The mean is not a resistant measure of center.

2. The median is not pulled into the direction of the most extreme observations. The median is not sensitive to extremes, i.e. the median is a resistant measure of center.

3. When the data is skewed, therefore, the median is the preferred measure of center.

4. The mode may not be near the center and, thus not useful as a measure of center.

Relative positions of the mean and median for (a) right-skewed, (b) symmetric, and (c) left-skewed distributions

Section 3.2

Measures of Variation

Five starting players on two basketball teams

Example:

Shortest and tallest starting players on the teamsExample:

Definition 3.5

Range of a Data Set

The range of a data set is given by the formula

Range = Max – Min,

where Max and Min denote the maximum and minimum observations, respectively.

∑

∑ N = 1+2+3+4+5+6+7+8+9+10 N = 1

10

∑ N = ? N = 1

10

Definition 3.6

Five starting players on basketball Team I.

Example:

Five starting players on basketball Team I

Example:

Five starting players on basketball Team I

Example:

Five starting players on basketball Team II

Example:

Five starting players on basketball Team 1.

Example:

Five starting players on basketball Team 1.

Example:

Formula

Summary:

• Distribution shapes

• Descriptive Measures => Central Measures => Mean, Median, Mode.

• Measures of Variations => Standard Deviation

Next ...

• Lab : Finish section 2.3 and Quiz 1 (1.1-2.3)

• Sections: 3.3 & 3.4

Thank You