Descriptive Statistics - Basic Statistics for Sociology - Lecture Slides, Slides of Statistics for Psychologists

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Descriptive Statistics

Healey Chapters 3 and 4

(2nd^ Cdn Ch. 3)

Measures of Central Tendency And Dispersion

Measures of Central Tendency

• 1. Mode = can be used for any kind of data but only measure of central tendency for nominal or qualitative data.
• Formula: value that occurs most often or the category or interval with highest frequency.
• Note: Omit Formula 3.1 Variation Ratio in Healey and Prus 2 nd^ Cdn.

Central Tendency (cont.)

• 2. Median = exact centre or middle of ordered data. The 50th percentile.
• Formula:
• Array data.
• When sample even #, median falls halfway between two middle numbers.
• To calculate: find(n/2)and (n/2)+1, and divide the total by 2 to find the exact median.
• When sample is odd #, median is exact middle (n+1) /2)

Example for Raw Data:

• Suppose you have the following set of test scores:
• 66, 89, 41, 98, 76, 77, 69, 60, 60, 66, 69, 66, 98, 52, 74, 66, 89, 95, 66, 69
• 1. Array data:
• 98 98 95 89 89 77 76 74 69 69 69 66 66 66 66 66 60 60 52 41 N = 20 (N is even)

Median for Aggregate (grouped) Data

• This formula is shown in Healey 1 st^ Cdn Edition and in Healey 8e but NOT in 2 nd^ Cdn
• We will NOT COVER this one!

Properties of median :

• • for numerical data at interval or ordinal level
• -"balance point“
• -not affected by outliers
• -median is appropriate when distribution is highly skewed.

Example for Mean

• Formula: = ΣXi / N

= 1446 / 20 = 72.

The mean for these test scores is 72.

Mean for Aggregate (Grouped) Data

(Note: 1st^ Cdn. Edition: use this formula!

Omitted in 2 nd^ Cdn. Ed. but covered in class)

• To calculate the mean for grouped data, you need a frequency table that includes a column for the midpoints, for the product of the frequencies times the midpoints (fm).

Formula: = Σ (fm) N

Calculating Mean for Grouped

Data:

Formula: = Σ (fm)

N = 1420 / 20 = 71

The mean for the grouped data is 71.

Properties of the Mean:

**- only for numerical data at interval level

• "balance point“
• can be affected by outliers = skewed distribution
• tail becomes elongated and the mean is pulled in direction of outlier.**

Example… no outlier: $30000, 30000, 35000, 25000, 30000 then mean = $ but if outlier is present, then: $130000, 30000, 35000, 25000, 30000 then mean = $ (the mean is pulled up or down in the direction of the outlier)

Measures of Dispersion

• Describe how variable the data are.
• i.e. how spread out around the mean
• Also called measures of variation or variability

Variability for Non-numerical Data

(Nominal or Ordinal Level Data)

• Measures of variability for non-numerical nominal or ordinal) data are rarely used
• We will not be covering these in class
• Omit Formula 4.1 IQV in Healey and Prus 1st Canadian Edition and in Healey 8e
• Omit Formula 3.1 Variation Ratio in Healey and Prus 2nd^ Canadian Edition

Interquartile Range (Q):

• This is the difference between the 75th and the 25th percentiles (the middle 50%)
• Gives better idea than range of what the middle of the distribution looks like.

Formula: Q = Q 3 - Q 1 (where Q 3 = N x .75, and Q 1 = N x .25) Using above data: Q = Q 3 - Q 1 = (6th^ – 2nd^ case) = $30000-25000 =$ The interquartile range (Q) is $5000.

3. Variance and Standard Deviation:

• For raw data at the interval/ratio level.
• Most common measure of variation.
• The numerator in the formula is known as the sum of squares , and the denominator is either the population size N or the sample size n-
• The variance is denoted by S 2 and the standard deviation, which is the square root of the variance, by S