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Social Statistics: Mean, Median, and Mode. This week. Mean Median Mode. How do we decide which is “best”?. The overall goal of central tendency is to find the single score that is most representative for the distribution. Measures of Central Tendency. Mean: Arithmetic average - PowerPoint PPT Presentation
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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