# Descriptive (Univariate) Statistics Percentages (frequencies) Ratios and Rates Measures of Central Tendency Measures of Variability Descriptive statistics.

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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 MeanSum of all values / N MeanSum of all values / N MedianCenter of the distribution (case that cuts the MedianCenter of the distribution (case that cuts the sample into two) ModeMost frequently occurring value ModeMost 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  RangeThe 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 Descriptive Statistics HIGHEST YEAR OF SCHOOL COMPLETED Minimum Maximum Mean SDVariance 02013.262.8698.232 N 2808

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