Descriptive Statistics: Maarten Buis Lecture 1: Central tendency, scales of measurement, and shapes of distributions.

Descriptive Statistics:Maarten Buis

Lecture 1:

Central tendency, scales of measurement,

and shapes of distributions

Outline

• Practicalities

• Central tendency

• Scales of measurement

• Shapes of distributions

Two statistics courses

• Descriptive Statistics (McCall, part 1)

• Inferential Statistics (McCall, part 2)

Course Material• McCall: Fundamental Statistics for

Behavioral Sciences.

• SPSS (available from Surfspot.nl) and chapter 2 of Field

• Lectures: 2 x a week

• computer labs: 1 x a week.

• course mailing list: [email protected]

• course website

setup of lectures

• Recap of material assumed to be known

• New Material

• Student Recap

How to pass this course

• Read assigned portions of McCall before each lecture• Do the exercises• Do the computer lab assignments, and hand them in

before Tuesday 17:00!• come to the computer lab• come to the lectures• ask questions: during class or to the course mailing

list• answer questions

Recap: mean, median, mode

• Mean of 1, 1, 2, 4 is (1 + 1 + 2 + 4)/4 = 2

• Median of 1, 1, 2, 4 is the middle observation, here two middle observations: 1 and 2. Use mean of middle observations, which is 1.5.

• Mode of 1, 1, 2, 4 is the most common value: 1.

Recap: quadratic

• 12 = 1 x 1 =1

• 22 = 2 x 2 = 4

• 32 = 3 x 3 = 9

• squaring makes large numbers much larger than small numbers

Recap: absolute value

• |3| = 3

• |-3| = 3

• just loose the minus sign if it is there

Data: rents of rooms

rent rent

room 1 175 room 11 240

room 2 180 room 12 250

room 3 185 room 13 250

room 4 190 room 14 280

room 5 200 room 15 300

room 6 210 room 16 300

room 7 210 room 17 310

room 8 210 room 18 325

room 9 230 room 19 620

room 10 240

What is a reasonable summary

• One always makes errors

• What if you choose that number that minimizes the sum of the absolute errors?

• If you want to put more weight on preventing large errors you could minimize the sum of the squared errors

mean and median

• mean is that summary that minimizes the sum of the squared errors

• median is that summary that minimizes the sum of the absolute errors

Measurement

• assigning numbers to observations: for example rents to rooms

• scale of measurement:– nominal– ordinal– interval/ratio

nominal

• == assigning numbers to classify observations in categories

• The categories are exclusive, but have no further relationship with one another.

• typical example: religion

ordinal

• == assigning numbers with the purpose to order observation

• It is meaningful to speak of more or less, or lower or higher

• typical example: education

Interval

• == assigning numbers to compare differences

• It is meaningful to say that the “distance between A and C is larger than between B and C”

• Typical example: temperature, intelligence• Hard to find really good examples, often

combined with ratio

ratio

• == assigning numbers to compare ratios of observations

• requires an absolute zero point

• It is meaningful to say “A is twice B”

• typical examples: age, income, percentage immigrant children in a classroom

What is the scale of measurement of:

• Choice of Party during an election

• Gender

• exam grades

• highest achieved level of education: primary, secondary, or tertiary

what is the scale of measurement of :

• income

• percentile of income (top 5% or bottom 20%)

• highest level of education in years

Why bother?

• Determines which statistical techniques are meaningful: mean religion or most common religion

• Use common sense

Central Tendency 2

• Nominal Mode

• Ordinal Mode or Median

• Interval/ratio Mode, Median, or Mean

Dichotomous variable

• only two answers possible: yes/no, male/female, 1/0

• Every variable can be dichotomized

• Dichotomous variables can be treated as interval variables: mean is meaningful: percentage “yes”.

Frequency distribution

• A frequency distribution shows how many times a value occurs within a variable

• Can be visualized in a histogram, frequency polygon, pie chart

rent Freq. Percent Cum.

175 1 5,26% 5,26%

180 1 5,26% 10,53%

185 1 5,26% 15,79%

190 1 5,26% 21,05%

200 1 5,26% 26,32%

210 3 15,79% 42,11%

230 1 5,26% 47,37%

240 2 10,53% 57,89%

250 2 10,53% 68,42%

280 1 5,26% 73,68%

300 2 10,53% 84,21%

310 1 5,26% 89,47%

325 1 5,26% 94,74%

620 1 5,26% 100,00%

Total 19 100.00%  

Shapes of distribution

• The mean, median and mode are equal in unimodal symmetric distributions.

• The mean and median are equal in multimodal symmetric distributions

• Skewness, in a right skewed distribution the mean is right of the median, the income distribution is an example of a right skewed distribution.

Shapes of distributions

• Kurtosis: leptokurtotic (flat) or platycurtotic (peaked)

• Uniform distribution, each value is equally likely

• For the fans: Mean, variance, skewness, and kurtosis are the first four moments of a distribution (p. 49 McCall)

Do before Wednesday

• Read:– McCall Ch 1: 6-14– McCall Ch 2: entirely– McCall Ch 3: 54-63

• Exercises:– 1.1-1.3– 2.1-2.7, 2.11, 2.13

Student recap