measuring and describing variables pt 2

frequency distribution

a tabular summary of a variable’s values

• commonly used in data presentations of all kinds

  • survey research and journalistic polls to marketing studies and corporate annual reports

raw frequency

count of individuals at each of the variable’s values

total frequency

total at the bottom of the column

what kind of charts are frequency distributions often in the form of?

bar chart

what do charts help us with?

help us communicate the most important features of data effectively

what kind of charts do we wanna avoid?

pie charts

the greatest amount of dispersion occurs when _____

cases are equally spread among all values of the variable

proportion

raw frequency divided by the total frequency

percentage

proportion multiplied by 100

how do we organize a frequency distribution table with ordinal values?

the order of the rows in a frequency distribution table of bars in a bar chart must be consistent with the relative rank of a variable’s values

cumulative percentage

records the percentage of cases at or below any given value of the variable

what can we do with the cumulative percentage?

we can locate the median, the value of the variable that (as closely as possible) divides the cases into two equal-sized groups

percentile

reports the percentage of cases in a distribution that lie below it

• serves to locate the position of an individual value relative to all other values

quantile

specific value in a data distribution that divides the data into equal portions or groups

• ex: quartiles, which divides data into four equal groups

bimodal distribution

frequency distribution having two different values that are heavily populated with cases

• two modes are separated by more than one nonmodal category

• we would not want to use a single mode to describe the central tendency of this distribution

non-parametric statistics

analyzing non-numerical data

• provide robust and flexible tools for data analysis

• work w/ any type of data, but less powerful than parametric statistics

• work without making strong assumptions about the data and are often used when the data don’t follow specific patterns or distributions

parametric stats

describe and summarize quantitative data

• based on assumptions about the distribution of the variable’s values

what columns does frequency distributions have?

frequencies, percentages, and cumulative percentages

histograms

another method of graphing the distribution of an interval-level variable with many unique values

• shows the percentage or frequency of cases falling into intervals of the variable

density plots

alternative to histograms for visualizing the distribution of an interval-level variable

• display a “running average” of observations across the range of observed values

• allow researchers to “zoom in” to show greater detail and “zoom out” to reveal general patterns in data

another way to describe the median

50th percentile value; the value that divides observations into equal-sized groups

name the three ways to describe the dispersion of an interval variable

range, interquartile range, and standard deviation

range

the maximum actual value minus the minimum actual value (largest - smallest)

interquartile range (IR)

the range of a variable’s values that defines the “middle half” of a distribution—the range between the upper boundary of the lowest quartile (which is the same as the 25th percentile) and the lower boundary of the upper quartile (the 75th percentile)

standard deviation

summarizes the extent to which the cases in an interval-level distribution fall on or close to the mean of the distribution

• in gauging variation in interval-level variables, standard deviation is the measure of choice

• measures the typical amount of deviation of a variable’s values from its mean value

• although it is a more precise measure of dispersion than those applied to nominal and ordinal variables, standard deviation is based on the same general principles

how to calculate the standard deviation

1. calculate each value’s deviation from the mean

2. square each deviation

3. sum the squared deviations

4. divide the sum of the squared deviations by n - 1 to find the variance

5. take the square root of the variance

skewness

measure of symmetry: the more skewed the distribution, the less symmetrical it is

• can be a positive or negative number

• distributions with a longer, or skinnier, right-hand tail have a positive skew

• those with a skinnier left-hand tail have a negative skew

kurtosis

measures the shape of a distribution, specifically how much it deviates from a bell-curve distribution

• provides information about the tails and peaks of a distribution and the number of extreme values observed

• always a positive number

leptokurtic

if a variable’s kurtosis > 3

• there are more values in the tails of the distribution, which indicates greater variability and the potential for more rare or extreme events

mesokurtic

if a variable’s kurtosis = 3

• distribution closely resembles a bell-shaped curve with a moderate amount of variability

platykurtic

if the variable’s kurtosis < 3

• distribution has a relatively flat peak and light tails, suggesting less variability and fewer extreme values

excess kurtosis

kurtosis - 3

• make it easier to classify the distribution as leptokurtic, mesokurtic, or platykurtic

box plot

communicates a five-number summary of a variable:

• minimum value, lower quartile (25th percentile), median, upper quartile (75th percentile), and maximum value

resistant measure of central tendency

extreme values may have an obvious effect on the mean, but they have little effect on the median

• the median is impervious to the amount of variation in a variable

• the median reports the value that divides the respondents into equal-sized groups, unfazed by the distribution’s skew