Measurements Statistics WEEK 6

Lesson Objectives Review Descriptive Statistics Descriptive / Survey Level of measurements Descriptive Statistics

Looking at descriptive survey Define objectives Define resources available Identify study population Identify variables to study Develop instrument (questionnaire) Create sampling frame Select sample Pilot data collection Collect data Analyse data Communicate results Use results

Levels of measurements Quantitative and Qualitative variables Quantitative variables are measured on an ordinal, interval, ratio scale and nominal scale. If five-year old subjects were asked to name their favorite color, then the variable would be qualitative. If the time it took them to respond were measured, then the variable would be quantitative

Ordinal Measurements with ordinal scales are ordered in the sense that higher numbers represent higher values. The intervals between the numbers are not necessarily equal. For example, on a five-point rating scale measuring attitudes toward gun control, the difference between a rating of 2 and a rating of 3 may not represent the same difference as the difference between a rating of 4 and a rating of 5. There is no "true" zero point for ordinal scales since the zero point is chosen arbitrarily. The lowest point on the rating scale is usually chosen to be 1. It could just as well have been 0 or -5.

Interval scale On interval measurement scales, one unit on the scale represents the same magnitude on the trait or characteristic being measured across the whole range of the scale. For example, if anxiety were measured on an interval scale, then a difference between a score of 10 and a score of 11 would represent the same difference in anxiety as would a difference between a score of 50 and a score of 51.

Interval scale Interval scales do not have a "true" zero point, however, and therefore it is not possible to make statements about how many times higher one score is than another. For the anxiety scale, it would not be valid to say that a person with a score of 30 was twice as anxious as a person with a score of 15. A good example of an interval scale is the Fahrenheit scale for temperature. Equal differences on this scale represent equal differences in temperature, but a temperature of 30 degrees is not twice as warm as one of 15 degrees

Ratio scale Ratio scales are like interval scales except they have true zero points. A good example is the Kelvin scale of temperature. This scale has an absolute zero. Thus, a temperature of 300 Kelvin is twice as high as a temperature of 150 Kelvin

Nominal scale Nominal measurement consists of assigning items to groups or categories. No quantitative information is conveyed and no ordering of the items is implied. Nominal scales are therefore qualitative rather than quantitative. Religious preference, race, and sex are all examples of nominal scales. Frequency distributions are usually used to analyze data measured on a nominal scale. The main statistic computed is the mode. Variables measured on a nominal scale are often referred to as categorical or qualitative variables.

Categorizing data Discrete data: finite options (e.g., labels) Gender Female 1 Male 2 Discrete: nominal, ordinal, interval Continuous data: infinite options Test scores 12 18 23.5 Continuous: ratio Discrete data is generally only whole numbers, whilst continuous data can have many decimals

Descriptive vs. Inferential Statistics

Descriptive vs. Inferential Statistics Used to summarize a collection of data in a clear and understandable way Inferential Used to draw inferences about a population from a sample “generalize to a larger population” Common methods used Estimation Hypothesis testing

Descriptive Statistics

Mean and standard deviation Central Tendency Measures the location of the middle or the center of the Mean - Average Median: Centre of the distribution Mode : Most frequently occurring score in a distribution Standard Deviation Measure of spread

Levels and measures

Describing nominal data Nominal data consist of labels e. g 1 = no, 2 = yes Describe frequencies Most frequent Least frequent Percentages Bar graphs

Frequencies No. of individuals obtaining each score on a variable Frequency tables Graphically ( bar chart, pie chart) Also %

Displaying data for gender

Mode Most common score Suitable for all types of data including nominal Example: Test scores: 16, 18, 19, 18, 22, 20, 28, 18

Describing ordinal data Data shows order e.g ranks Descriptives frequencies, mode Median Min, max Display Bar graph Stem and leaf

Example: Stem and Leaf Plot Underused. Powerful Efficient – e.g., they contain all the data succintly – others could use the data in a stem & leaf plot to do further analysis Visual and mathematical: As well as containing all the data, the stem & leaf plot presents a powerful, recognizable visual of the data, akin to a bar graph.

Example The data: Math test scores out of 50 points:  35, 36, 38, 40, 42, 42, 44, 45, 45, 47, 48, 49, 50, 50, 50. Separate each number into a stem and a leaf.  Since these are two digit numbers, the tens digit is the stem and the units digit is the leaf.   The number 38 would be represented as Stem 3 Leaf 8 Group the numbers with the same stems.  List the stems in numerical order.  (If your leaf values are not in increasing order, order them now.)  Title the graph To find the median in a stem-and-leaf plot, count off half the total number of leaves.

Describing interval data Interval data are discrete but also treated as ratio/continuous Descriptives Mode Median Min, max Mean if treated as continuous

Distribution Describing Mean Average, central tendency Deviation Variance Standard deviation Dispersion If the bell-shaped curve is steep, the standard deviation is small. When the data are spread apart and the bell curve is relatively flat, you have a relatively large standard deviation

Distribution Describing Skewness a measure of symmetry, or more precisely, the lack of symmetry Lean, tail +ve : tail at the right

Distribution Describing Kurtosis Flatness/peakedness of distribution + ve : peaked data sets with high kurtosis tend to have a distinct peak near the mean, decline rather rapidly, and have heavy tails. data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak

Approx. same skewness, different kurtosis

Describing Ratio Data Can talk meaningfully about ratio data Measures - central tendency, dispersion

Describing Ratio Data Displaying frequency

Using SPSS Next