Central Tendency

ordinal scale

-a lot of researchers believe that it is not good to use the mean for ordinal data -the median is perferred (defined by direction) -allow you to determine direction (greater than or less then) but do not allow you to determine distance

discrete variables

-mode use -exist only in whole, indivisible categories...often are numerical values, such as number of children in a family or number of rooms in a hosue -when they produce numerical scores, it is possible to calculate the means -the calculated means are usually fractional values that cannot actually exist ...mode is the best in this case because whole number values by the mode

nominal scales

-mode use -measured by name, do not measure distance, direction, or quantity -it is impossible to compute a mean or median for data from a nominal scale

describing shape

-mode use -mode gives an indication of the shape of the distribution as well as a measure of central tendency

how do you deicide which central tendency to use?

-th median is commonly used when reporting the average value for a skewed distribution -ex: the distribution of personal incomes is very skweed, with a small segment of the population earning incomes that are astronomical...the extreme values distort the mean so the median is perferred measure of central tendency

rules or bar graphs

-the height of a graph should be approximately 2/3 to three quarters of its length -normally you start numbering both the x-axis and the y-axis with zero at the point where the two axes intersect. Howecer, when a value of zero is part of the data, it is common to move the zero point away from the intersection so that the graph does not overlap the axes

horizontal axis

-when preparing a graph, it is customary to list the different groups or treatment conditions on the horizontal axis (typically, these are the different values that make up the independent variable or quasi-indepedent varialbe)

bimodal

2 modes

1 inch =

2.54 centimeters

open-ended distributions

a distribution is said to be open-ended when there is no upper limit (or lower limit) -cant get the mean but you can get the median

introducing a new score or removing a score

adding a new score to a distribuation, or removing an existing score, usually chnages the mean -the exception is when the new score (or the removed score) is exactly equal to the mean

the mean as a blance point

alternative describes the mean as a balance point for the distribution.

dividing the total equally

alternative is to think of the mean as the amount each individual receives when the total (X) is dividied equally among all of the individuals (N) in the distribution -used in problems when you know the mean and must find the total

means or medians in grphs

are displayed using a line, histogram, or bar graph depending on the scale of measurement used for the indepdent variable

negatiely skewed distributions

are lopsided in the opposite direction, with the scores piling up o the right-hand side and the tail tapering off to the left -the mode is on the right-hand side (with the peak), whereas the mean is displaced toward the left by the extreme scores in the tail -the median is located between the mean and the mode - in order from smallest value to largest value (left to right), the 3 measures of central tendency for a negatively skewed distribution are the mean, he median, and the mode

bar graphs

are used to present means or median when the groups ortreatments showen on the horizontal axis are measured by a nominal or an ordinal scale

graphs

can be used to report and compare measures of central tendency -to display values obtain for sample means, but occasionally sample median are reported in graphs -the value of a graph is that it allows several means to be shown simutaneously, so it is possible to make quick comparisions between groups or treatment conditions

characteristics of the mean (changing a score)

changing the value of any score chnges the mean.

the weighted mean

combining two sets and scores and getting the overall mean x1+x2/n1+n2

finding the median for most distributions

defining the median as the midpoint of a distribution means that the scores are divided into two equal-sized groups -to find the median list the scores in order from smallest to largest -the median is the first point you reach that is greater than 50% of the scores of the distribution -the median can be = to a score in the list or it can be a point between two scores

skewed distributions

especially distributions for continuous variables, there is a strong tendency for the mean, median and mode to be located in predictably different positions -to have exactly 50% of the distribution on each side, the emdian must be located to the right of the mode -the mean is located to the right of the median because it is the measure influenced most by the extremem scores in the tail and is displaced farthest to the right toward the tail of the distrubtion -the order of the three measures of central tendency from smallest to largest (left to right) is the mode, the median and the mean

mean

for a distribution is the sum of the scores divided by the number of scores

finding the precise median for a continuous variable

fraction=number needed to reach 505%/number in the interval -same as the 50th percentile -dividing scores into fractional parts

a bimodal distribution that is summetrical

has the mean and median together in the center with the modes on each side

central tendency

identifies the average, or typical, score to serve as a representative value for each group -then we can use the two averages to describe the two groups and to measure the difference between them -the concept of an average, or representative score is called central tendency

adding or subtracting a constant from each score

if a constant value is added to every score in a distribution, the same constant is added to the mean. if you subtract a constant from every score, the same constant is subtracted from the mean -adding a constant to every score is important

multiplying or dividing each score by a constant

if every score in a distribution is multiplied by (or divided by) a constant value, the mean hanges in the same way -it is a common way of changing the unit of measurement

mode

in a frequency distribution, the mode is the score or category that has the greatest frequency -no symbols to show it -defnition for mode is the same for a population and for a sample distirbution -to determine the typical or average value for any scale of measurement -the mode is the only measure of central tendency that can be used with the data from a nominal scale of measurement -in a distribution it is possible to have more than one mode -if lots a modes there are said to be no modes? -identifies the location of the peak or peaks in the frequency distribution graph

def. central tendency

is a statistical measure to determine a single score that defines the center of a distribution. the goal of central tendency is to find the single score that is most typical or most representative of the entire group -single most representative score

term middle

is defined by the number of scores

taller peak

major mode

3 different methods for measuring central tendency

mean, median, and mode because it can be difficult to find the center

shorter peak

minor mode

a rectangular distribution has no

mode because all x values occur with the same frequency. still, the mean and the median are in the center of the distribution

multimodal

more than 2 modes

to change a set of measurements from minutes to seconds. you

multply by 60

u

population mean

M

sample mean

the median

second measure of cnetral tendency -goal of the median is to locate the midpoint of the distribution -there are no specific symbols or notation to identify the median...just by the wordmedian -if the scores in a distribution are listed in order from smallest to largest, the median is the midpoint of the list. More specfically, the median is the point on the measurement scale below which 50% of the scores in the distribution are located

undetermined values

sometimes, you get a situation in which an individual has an unkown or undetermined score...ften in learning experiments in which you are measuring the number of errors (or amount of time) -can get the median but hard to get the mean page 91

the mean is usually

the perferred measure of central tendency

symmetrical distributions

the right-hand side of the graph is a mirror image of the left-hand side -if the distribution is perfectly symmetrical, the median is exactly at the center because exactly half of the areaa in the graph is on either side of the center -the mean is also exactly at the center -the mean and median are the same -if the distribution is roughly symmetrical but not perfect, the mean and median are close together in the center of the distribution -if a symmetrical distrubtion only has 1 mode, it is also in the center of the distribution -perfectly symmetrical, all 3 tendencies are the same

vertical axis

values for the dependent variable (The scores) are listed on the vertical axis

extremem scores or skewed distributions

when a distribution has a few extremem scores, scores that are very different in value from most of the others, then the mean may not be a good representative of the majority of the distribution

to change from inches to feet

you divide by 12