Chapter 3: Data Description

3-1 Intro & 3-2 Measures of Central Tendency

Avg American male: 5’9” Avg American female: 5’4” Sick in Bed 7 days/yr, misses 5 days of work On the average day, 24 million people are

bit by an animal By 70th birthday, avg person has eaten 14

steers, 1,050 chickens, 3.5 lambs, and 25.2 hogs

Some Data about being “Average”

Chapter 3 will show us how to obtain and interpret descriptive or traditional statistics such as:◦ measures of average/central tendency-mean,

median, mode, midrange◦ measures of variation/spread/dispersion-range,

variance, standard deviation◦ measures of position (tell where a specific data

value falls within the data set or its relative position in comparison with other data values)-percentiles, deciles, quartiles

3-1: Introduction

Exploratory Data Analysis: used to explore data to see what they show◦ Boxplot◦ Five-number summary◦ Different from traditional techniques which are

used to confirm conjectures about the data.

3-1: Introduction

Objective: To summarize data using measures of central tendency such as the mean, median, mode, and midrange.

3-2: Measures of Central Tendency

statistic: characteristic or measure obtained by using the data values from a sample

parameter: characteristic or measure obtained by using all the data values from a specific population.

3-2: Measures of Central Tendency

Statistically-don’t round until the last step.

Examples in book-not practical to show long decimals in the intermediate calculations. Final answer shown will not have rounded until the last step.

General Rounding Rule

Sum of numbers divided by total number of values.

Σ: Greek capital letter “sigma” which means to find the sum of

Σ X means to find the sum of the x values in the data set

Mean (arithmetic average)

Mean: where n represents the total number of values in the sample.

Mean: where N represents the total number of values in the population.

Mean, cont’dn

xxxxx n ...321

Nxxxx N ...321

Round the mean to one more decimal place than your data.

Rounding Rule for the Mean

Finding the mean

Examples 3-1, 3-2 on p.98-99

Example

Make a table with the following columns: A-Class, B-Frequency (f), C-Midpoint (Xm)

and D - f·Xm.

Find the midpoints of each class and place them in column C

Multiply the frequency by the midpoint for each class and place the product in column D.

Divide the sum from column D by the sum of the frequencies from column B.

Finding the Mean for Grouped Data

Finding the mean from a frequency distribution

Example 3-3 on p.99-100

Example

When a data set is ordered it’s called a data array.

The median is the midpoint of the data array. The symbol is MD.

To find the median:◦ Arrange data in order from least to greatest.◦ Select the middle number.◦ If two numbers occur in the middle, find their

average.

Median

3-4 thru 3-8 on p.101-103

Median Examples

The value that occurs most often in a data set is called the mode.◦ Only one mode in a set: unimodal◦ Two modes in a set: bimodal◦ More than two modes: multimodal◦ No modes: no mode

Mode for grouped data: Called the modal class. It is the class with the highest frequency.

Mode

Examples 3-9 thru 3-11 on p.103-104

Modal Class Example 3-12 on p.104

Mode Examples

Ex 3-14 on p.105

Differences in Measures

The midrange is defined as the sum of the lowest and highest values in the data set, divided by 2. Symbol: MR

Midrange

Find this when all values are not equally represented.

Weighted Mean

Last week, 3 van drivers for company ABC filled their gas tanks. Driver 1 bought 14 gallons at $3.95/gal; driver 2- 25 gallons at $3.49/gal; driver 33 - 5 gallons at $3.99/gal.

What was the average price per gallon for ABC?

14(3.95)+25(3.49)+5(3.99)=162.50 14+25+5 = 44 162.50/44 = $3.69

Weighted Mean Example

Course Credits Grade

English I 3 A (4 points)Intro to Psych 3 C (2 points)

Biology I 4 B (3 points)Phys Ed 2 D (1 point)

Weighted Mean Example - GPA

Mean◦ Use all data values◦ Varies less than median or mode when using

samples from the same population.◦ Used in computing other statistics: variance◦ Unique for a data set and not necessarily one of

the data values◦ Can’t be computed when frequency distribution is

open-ended◦ Affected by extremely low or high values called

outliers, thus it might not be the appropriate average for certain situations

Properties and Uses of Central Tendency

Median◦ Used when need to find middle or center of data

set◦ Used when need to see if data value falls into

upper half or lower half of distribution◦ Used for open-ended distribution◦ Affected less than mean by outliers

Properties and Uses of Central Tendency

Mode◦ Used when most typical case is desired◦ Easiest average to compute◦ Can be used when data are nominal◦ Not always unique

Properties and Uses of Central Tendency

Midrange◦ Easy to compute◦ Gives the midpoint◦ Affected by outliers

Properties and Uses of Central Tendency

Three most important shapes: positively skewed, symmetric, negatively skewed◦ Positively skewed (right-skewed):

majority of data falls to the left, “tail” is to the right, Mean is to the right of the median Mode is to the left of the median

◦ Symmetric: data are evenly distributed on both sides of the mean. When unimodal: mean, median, mode are the same

and located at the center of the distribution

Distribution Shapes

Negatively skewed (left-skewed): majority of data falls to the right, “tail” is to the left, Mean is to the left of the median Mode is to the right of the median

Distribution Shapes, cont’d

p.109-110 #1-6

Applying the Concepts

p.110-111 #1, 3, 13, 15, 27, 29, 31-34

Assignment: