Download Descriptive Statistics

Descriptive (Univariate) Statistics
Percentages (frequencies)
Ratios and Rates
Measures of Central Tendency
Measures of Variability
Descriptive statistics describes
variables—We are not testing
relationships between variables
Central Tendency


What is the average value of a variable in a
range of values for a given population?
MEASURES of Central Tendency
 Mean Sum of all values / N
 Median Center of the distribution (case that
cuts the
sample into two)
 Mode Most frequently occurring value
Calculating Each Measure
Mean = ∑X / n
– (Sum of all values divided by the sample size).
Mode = Count the most frequently occurring
value.
Median
Median = Odd # of cases (Md = middle value)
Finding the Middle Position:
(n + 1) / 2 (= position of the middle value)
Example:
11, 12, 13, 16, 17, 20, 25 (N=7)
Md = (7+1) / 2 = 4th value = 16
50% of cases lie above and below 16
Median Continued
Median = even # of cases
There will be two middle cases
Md. = the average of the scores of the two middle
cases.
Example:
11, 12, 13, 16, 17, 20, 25, 26
Position of the middle value = (8+1)/2 = 4.5
Md = 16 + 17 (two middle cases) / 2 = 16.5
NOTE = Need to sort your values before locating the
Md
Why might we use the median instead
of the mean?
Skewed Distributions
See board
Mean is most sensitive to outliers
EXAMPLE:
5, 6, 6, 7, 8, 9, 10, 10 Md. 7.5 Mean 7.63
5, 6, 6, 7, 8, 9, 10, 95 Md. 7.5 Mean 18.25
Measures of Variability
Variability—scatter of scores around the mean.
How do scores cluster around the mean?
Example: Say the average price of a home in
Bakersfield is (say 150,000). Can you buy a
home in Hagen Oaks for 150,000?
See bell curve (mean = 150K, Sd = 10K)
Measures of Variability
 Range The distance between the highest
and lowest score (subtract the
lowest value from the highest
value)
 A rough measure.
Standard Deviation
Deviation = The distance of a given raw
score from the mean (X – Mean).
We need a summary measure that accounts
for all of the scores in a distribution.
Variance and SD are summary measures
Calculate the SD by taking the Square Root of
Variance
Variance
Variance = ∑ (X-mean) squared/n
Dividing by n controls for the number of scores
involved.
SD = Square root of variance
We take the square root of variance b/c it is easier
to interpret.
Spread Around the Mean
Theoretically:
 34.13% of the cases fall 1 SD above & 1 SD
below the mean.
 47.72% fall 2 SDs above mean & 2 SDs below
the mean.
 49.87% of cases fall 3 SDs above & 3 SDs
below the mean.
Housing Cost Example Cont.
If the mean is 150,000 & Sd is 10,000 then:
 99.74% of the cases fall between 120,000
(3 SDs below the mean) & 180,000 (3 SDs
above the mean)
Levels of Measurement & Descriptive
Statistics
Nominal
 Frequency Distribution
 Modal Category
Ordinal
 Frequency Dist.
 Modal Category
 Mean in some cases (i.e. a scale)
Interval/Ratio
 Mean, Md., Mode
 Variance & Standard Deviation
Practice Interpretation
Descriptive Statistics
HIGHEST YEAR OF SCHOOL COMPLETED
Minimum Maximum
Mean
SD
0
13.26
2.869
N 2808
20
Variance
8.232