EDUC 7741
Quantitative data analysis-descriptive statistics
Paris
Data Analysis: Descriptive Statistics
I.
II.
III.
What is “sadistics” ? Oops, I mean “statistics”
Distributions
A. Frequency distribution
B. Measures of central tendency
1. mean
2. median
3. mode
C. Measures of variability
1. Range
2. variance
3. standard deviation
D. Shapes of distributions
Standard Scores
A. Properties
B. Calculation of z scores
EDUC 7741
Quantitative data analysis-descriptive statistics
Paris
Data Analysis: Descriptive Statistics
Descriptive statistics- procedures and measures by which we describe a set of
quantitative data.
Scoring Data
Assigning numeric value to response categories. Similar scales should have same
scoring… p. 223 Creswell….may or may not want to include the # by the response
alternative…could influence responders. (5= desirable, higher value. Whereas 1 = less
desirable, lesser value)
Distributions- the group of all scores or observations on a variable. The shape of the
measured values of a variable within a sample or population. Based upon how many
members share the same values.
Frequency distribution is a listing of each possible score and the frequency of its
occurrence. (f)
Adequately describing a distribution involves providing information about its
location, dispersion and shape.
Measures of central tendency- (Creswell, p. 232, Table 8.3)
provide information about the location of a distribution on a scale of measurement.
Commonly referred to as “averages”. Describe what a “typical observation” in the
distribution is like.
Mean- arithmetic mean. Sum of all the scores divided by the number of scores.
Obtained by adding all the scores in the distribution and dividing by the total number of
scores.
Median- point of the scale of measurement below which one-half of the scores of
the distribution lie.
Mode- the score with the greatest frequency.
Measures of variability- indicates the dispersion of scores or the spread of scores around
the mean.
Range- number of units on the scale of measurement necessary to include the
highest and lowest scores of the data set. Provides little information about he pattern of
variation.
(Deviation- difference between an observed score and the mean of the distribution.)
Variance-the average of the squared deviations from the mean.
Standard deviation- square root of the variance. Both measures provide
information about the dispersion or shape of the distribution.
EDUC 7741
Quantitative data analysis-descriptive statistics
Paris
Measures of relative standing
Shapes of distributions- often researchers are more concerned with the assumed or
theoretical shape of distributions rather than their actual shape.
Normal distribution- Bell-shaped, more observations located around the average value,
fewer extremes as either end is approached. Often the basis for statistical analysis. Many
statistical tests make the assumption that the overall distribution of the variable under
study in the population is normally distributed.
Skewed positive- tilted so that the skinny tail goes to the right (towards + infinity).
Skewed negatively- tilted so that the skinny tail goes to the left (towards – infinity).
Leptokurtic- peaked (kangaroos LEAP)
Platykurtic- flattened- platy sounds like flatty or platypuses have big flat lips.
Uniform- looks like a rectangle; all values equally distributed across all members.
Bimodal- two values more popular that all others. Looks like two-humped camel.
Standard scores- (z- score) the number of standard deviations that an observed score is
located from the mean. Uses the standard deviation of the distribution as the unit of
measurement and then indicates the relative position of a single score in the entire
distribution of scores.
Properties
Shape of the distribution of z-scores is identical to that of the original distribution
Mean of the distribution of z-scores is zero, regardless of the value of the mean in
the original distribution.
Variance of the distribution of z-scores is 1, and so is the standard deviation.
To calculate-
X – xbar
z = ------s
X= observed score
X (xbar)= mean of observed distribution
s = standard deviation of the observed distribution
EXAMPLE: