EDU 660 Methods of Educational Research Descriptive Statistics John Wilson Ph.D.

Definitions  Quantitative data numbers representing counts or measurements

Definitions  Quantitative data numbers representing counts or measurements  Qualitative (or categorical or attribute) data can be separated into different categories that are distinguished by some non-numeric characteristics

Definitions  Quantitative data the incomes of college graduates

Definitions  Quantitative data the incomes of college graduates  Qualitative (or categorical or attribute) data the genders (male/female) of college graduates

 Discrete data result when the number of possible values is a ‘countable’ number 0, 1, 2, 3,... Definitions

 Discrete data result when the number is or a ‘countable’ number of possible values 0, 1, 2, 3,...  Continuous (numerical) data result from infinitely many possible values that correspond to some continuous scale Definitions 2 3

 Discrete The number of students in a classroom. Definitions

 Discrete The number of students in a classroom.  Continuous The value of all coins carried by the students in the classroom. Definitions

 nominal level of measurement characterized by data that consist of names, labels, or categories only. The data cannot be arranged in an ordering scheme (such as low to high) Example: Your car rental is a: Ford, Nissan, Honda, or Chevrolet Levels of Measurement of Data

 ordinal level of measurement involves data that may be arranged in some order, but differences between data values either cannot be determined or are meaningless. Example: Course grades A, B, C, D, or F. Your car rental is an: economy, compact, mid-size, or full-size car. Levels of Measurement of Data

 interval level of measurement like the ordinal level, with the additional property that the difference between any two data values is the same. However, there is no natural zero starting point (where none of the quantity is present) Example: The temperature outside is 5 degrees Celsius. Levels of Measurement of Data

 ratio level of measurement the interval level modified to include the natural zero starting point (where zero indicates that none of the quantity is present). For values at this level, differences and ratios are meaningful. Examples: Prices of textbooks. The Temperature outside is 278 degrees Kelvin. Levels of Measurement of Data

Levels of Measurement  Nominal - categories only  Ordinal - categories with some order  Interval – interval are the same, but no natural starting point  Ratio – intervals are the same, and a natural starting point

a value at the centre or middle of a data set Mean Median Mode Measures of the centre

Mean (Arithmetic Mean) AVERAGE The number obtained by adding the values and dividing the total by the number of values Definitions

Notation  denotes the addition of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x x

Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n  x x x N µ =  x x

Definitions  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude The Median is used to describe house prices in Toronto. Why not the Mean?

Definitions  Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data

a b c Examples  Mode is 5  Bimodal - 2 and 6  No Mode

Waiting Times of Bank Customers at Different Banks (in minutes) TD RBC

TD RBC TD RBC Mean Median Mode Midrange Waiting Times of Bank Customers at Different Banks in minutes

Measures of Variation Range Variance Standard Deviation

Measures of Variation Range value highest lowest value

Measures of Variation Variance Mean Squared Deviation from the Mean

(Root Mean Squared Deviation) Measures of Variation Standard Deviation

Population Standard Deviation Formula Root Mean Squared Deviation  ( x - x ) 2 N ==

Basketball Starting Line Height (inches)