Social Statistics: Mean, Median, and Mode

Statistical analysis involves many mathematical operations which depends on how our variables are measured Using number 1 to represent “Female”:

1 here is only the symbol. Using number 1 to represent the only

one child in the family: 1 here means real quantity.

Levels of measurement

Nominal: numbers or other symbols are assigned to a set of categories for the purpose of naming, labeling or classifying the observations. For example:1=female, 2=male Number here does not carry any

quantitative difference.

Levels of Measurement

Ordinal: numbers are assigned to rank-ordered categories ranging from low to high.

For example: upper class, middle class or working class We know that upper class is higher than

middle class But we do not know the magnitude of

differences between the categories, we do not know how much higher upper class is compared with the middle class

Levels of Measurement

Interval-ratio: If the categories (or values) of a variable can be rank-ordered, and if the measurements for all the cases are expressed in the same units,

Example: age, income, SAT scores We can compare values not only in terms of

which is larger or smaller but also in terms of how much larger or smaller one is compared with another.

Variables with a natural zero point are also called ratio variables.

Levels of Measurement

Variables that can be measured at the interval-ratio level of measurement can also be measured at the ordinal and nominal levels.

As a rule, properties can be measured at a higher level (interval-ratio is the highest) can also be measured at lower levels, but not vice versa.

Levels of Measurement

Levels of Measurement

Levels of Measurement

Several key social factors (gender, employment status, martial status) are dichotomies.

They are nominal

Dichotomous variables

Discrete vs. continuous variables Discrete: number of kids Continuous: Length or weights

Levels of Measurement

The number of people in your family Place of residence classified as urban, suburban,

or rural The percentage of university students who

attended public high school The rating of the overall quality of a textbook, on

a scale from “Excellent” to “Poor” The type of transportation a person takes to work Your annual income The U.S. unemployment rate The presidential candidate that the respondent

voted for in 2012

Levels of Measurement

The overall goal of central tendency is to find the single score that is most representative for the distribution.

How do we decide which is “best”?

Mean: Arithmetic average sum of scores divided by number of scores most frequently used it uses all scores in the set

Median: “Middle” score, when scores are in order

corresponds to the 50th percentile appropriate for skewed/open-ended distributions, and distributions with undetermined scores

Mode: Most frequently occurring (popular) score

appropriate for nominal data

Measures of Central Tendency

Mean

n

XX ___

• (x bar) : the mean• : sum of the data• : number of the data

The sample mean is the measure of central tendency which can approximate the population mean

The mean is very sensitive to extreme scores It can put the mean in some extreme

direction Make it less representative Less useful as a measure of central

tendency

Mean

Location Number of annual customers

Lanham Park Store 2150

Williamsburg Store 1534

Downtown Store 3564

Calculate mean

The mean or average number of shoppers in each store?

Using Excel to do that• use your own formula• use AVERAGE function

It is defined as the midpoint in a set of scores 50% of the scores fall above and one

half fall below.

Median

Odd number of data Rank them Median=middle one Example: 10, 9, 8, 7, 5 (median=8)

Even number of data Rank them Median= sum of two middle data/2 Example: 10, 9, 8, 7, 6, 5

(median=(8+7)/2=7.5)

Calculate median

The median is insensitive to extreme cases, where the mean is not.

To measure the central tendency: Have some extreme data, using median No extreme data, using mean Example: 14, 3, 2, 1, (mean=5,

median=2.5) Which represents better the central

tendency?

Median

Calculate the median of income level

Median in Excel

The mode is the value that occurs most frequently. Count the frequency of all the values in a

distribution The value that occurs most often is the

mode

Mode

Ten Most Common Foreign Languages Spoken in the United State, 2009

Calculate mode

Language Number of Speakers

Spanish 35,468,501

Chinese 2,600,150

Tagalog 1,513,734

French 1,305,503

Vietnamese 1,251,468

German 1,109,216

Korean 1,039,021

Russian 881,723

Arabic 845,396

Italian 753,992

Mode: Spanish

Listed are the weather conditions of 10 US cities on 11/14/2014. What is the mode?

Calculate mode

Chicago CloudyLos Angeles SunnyWashington DC Partly CloudyNew York CloudySeattle CloudySalt Lake City SnowBoston Partly CloudyPhoenix Mostly CloudyLexington Mostly CloudyNew Orleans Fair

Mean: No extreme scores and are not categorical

Median Extreme scores and you do not want to

distort the average Mode

Data are categorical in nature and values can only fit into one class

E.g. hair color, political affiliation, religion

When to use what

Input the table to Excel Select the data as Input

Range click Data Data Analysis in Data Analysis box, choose Descriptive Statistics tick “Labels in first row” Output Range=C1 tick “Summary statistics” click “OK”

Descriptive Statistics in Excel

Income Level

$135,456

$54,365

$37,668

$34,500

$32,456

$25,500

Descriptive Statistics

Calculate mean, median and mode for the following data:

Exercise 1

Score1 Score2 Score33 34 1547 54 1675 17 1324 26 1455 34 1546 25 1457 14 1138 24 1566 25 1545 23 123

Writing a sale report to your boss according to the figures of things sold today:

Exercise 2

Special Number Sold Cost

Huge Burger 20 $2.95

Baby Burger 18 $1.49

Chicken Littles 25 $3.50

Porker Burger 19 $2.95

Yummy Burger 17 $1.99

Coney Dog 20 $1.99

Calculate the average sale

Exercise 3

Toy July sale August Sale September Sale

slammer 12345.00 14453.00 15435.00

radar zinger 31454.00 34567.00 29678.00

lazertags 3253.00 3121.00 5131.00

Patient record Mean and median, which is better for

what?

Exercise 4

  12/1-12/7 12/8-12/15 12/16-12/23

0-4 years 12 14 15

5-9 years 15 12 14

10-14 years 12 24 21

15-19 years 38 12 19