Social Statistics: Mean, Median, and

Social Statistics: Mean, Median, and

Mode

Statistical analysis involves many mathematical operations which depends on how our variables are measuredUsing 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=maleNumber 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 classWe know that upper class is higher than middle classBut 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 scoresWe 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 variablesDiscrete: number of kidsContinuous: Length or weights

Levels of Measurement

The number of people in your familyPlace of residence classified as urban, suburban, or ruralThe percentage of university students who attended public high schoolThe rating of the overall quality of a textbook, on a scale from “Excellent” to “Poor”The type of transportation a person takes to workYour annual incomeThe U.S. unemployment rateThe 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 averagesum of scores divided by number of scoresmost frequently usedit uses all scores in the setMedian: “Middle” score, when scores are in ordercorresponds to the 50th percentileappropriate for skewed/open-ended distributions, anddistributions with undetermined scoresMode: Most frequently occurring (popular) scoreappropriate for nominal data

Measures of Central Tendency

Mean

(

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 meanThe mean is very sensitive to extreme scoresIt can put the mean in some extreme directionMake it less representativeLess useful as a measure of central tendency

Mean

LocationNumber of annual customersLanham Park Store2150Williamsburg Store1534Downtown Store3564

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 scores50% of the scores fall above and one half fall below.

Median

Odd number of dataRank themMedian=middle oneExample: 10, 9, 8, 7, 5 (median=8)Even number of dataRank themMedian= sum of two middle data/2Example: 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 medianNo extreme data, using meanExample: 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 distributionThe value that occurs most often is the mode

Mode

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

Calculate mode

LanguageNumber of SpeakersSpanish35,468,501Chinese2,600,150Tagalog1,513,734French1,305,503Vietnamese1,251,468German1,109,216Korean1,039,021Russian881,723Arabic845,396Italian753,992

Mode: Spanish

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

Calculate mode

Chicago Cloudy

Los Angeles Sunny

Washington DC Partly Cloudy

New York Cloudy

Seattle Cloudy

Salt Lake City Snow

Boston Partly Cloudy

Phoenix Mostly Cloudy

Lexington Mostly Cloudy

New Orleans Fair

Mean:No extreme scores and are not categoricalMedianExtreme scores and you do not want to distort the averageModeData are categorical in nature and values can only fit into one classE.g. hair color, political affiliation, religion

When to use what

Input the table to ExcelSelect 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

Score3

3

34

154

7

54

167

5

17

132

4

26

145

5

34

154

6

25

145

7

14

113

8

24

156

6

25

154

5

23

123

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

Exercise 2

S

pecial

Number Sold

C

ost

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

T

oy

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 recordMean 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