Measures of Central Tendency

descriptive statistics

tools that are used to organize and summarize data numerically

measures of central tendency

used to represent “central” or “typical” values in a dataset

what are the most common measures of central tendency

• mean

• median

• mode

what question do measures of central tendency help to answer

if i had to come up with one number that would best describe my dataset, what number would it be

mean

the arithmetic average of a set of data values

what equation does this solve for

mean

what does the greek letter sigma mean

add

what are greek letters typically associated with in statistics

parameters (truth)

population mean

the true average of a population

what does the greek letter mu mean

true mean of a population

what does a lowercase n indicate

total number of individuals in a sample

what does a capital N indicate

total number of individuals in a population

why does notation difference matter

it is the difference between a truth and an estimate

what does an x bar mean

estimated mean

what does this equation solve for

sample mean

what does this equation solve for

population mean

what are advantages of a mean

• easy to compute

• it is a unique descriptive (when you find a mean there is only one of them)

• it is the most representative measure of center because it takes every score value into account

what are disadvantages of a mean

it is sensitive to extreme scores and “outliers'“

outlier

a data value that is “far away” from the majority of other values in a data set

why does an outlier impact the quality of a summary

it can inflate or deflate the mean

median

the value that divides a dataset into 2 equal parts such that the same number of scores in the dataset fall above and below this value

steps to finding a median

• arrange all data values in order from least to greatest

• given an ODD number of data values, the median is the exact middle value in the ordered list

• given an EVEN number of data values, the median is the average of the middle 2 values in the ordered list

advantages of the median

because the median only looks at values in the middle of the dataset, it is less sensitive to extreme scores/outliers than the mean

disadvantage of the median

because it only looks at values in the middle of a dataset, the median is not as representative as a measure as the mean

mode

the score value that occurs most frequently in a data set

bimodal

if 2 scores have the same greatest frequency

multimodal

if more than 2 scores have the same greatest frequency

no mode

if no score value is repeated

advantage of the mode

it is the only measure of central tendency appropriate for use with qualitative data

disadvantage of the mode

it tends to be less informative than the mean or median when dealing with quantitative data

which central modes of tendency are preferred when working with quantitative data

mean and median

in the presence of an extreme score/outlier which mode of central tendency is most appropriate

the median

which measure of central tendency is the only one appropriate for use with qualitative data

mode

symmetric distributions

a distribution of data where the left half of its graph is a mirror image of the right half, and vice versa

example of symmetric distribution

bell shaped distribution

where is the mean/median/mode found in a symmetric distribution

exact center of the distribution, corresponds with peak

skewed distributions

distributions that are not symmetric

positively skewed distributions

a distribution that is not symmetric because its graph extends further to the right (long tail to the right)

in a positively scaled distribution what end do the majority of score values fall in

low end

in a positively skewed distribution what do the few outliers do to the distribution

extend it to the low end/right

in a positively scaled distribution are the outliers higher or lower

higher

in a positively skewed distribution, what is the order of measures of central tendency from least to greatest

mode<median<mean

in a positively skewed distribution why is the mean the greatest

because it hast to take the outliers into account which inflates it

negatively skewed distribution

a distribution that is not symmetric because its graph extends further to the left (it has a long tail to the left)

where do the majority of scores fall in a negatively scaled distribution

the high end

in a negatively skewed distribution what do the outliers do to the distribution

they pull the tail to the low end (left)

in a negatively skewed distribution are the outliers higher or lower

lower

in a negatively skewed distribution, what is the order of the central modes of tendency in order from least to greatest

mean<median<mode

why is the mean the lowest in a negatively skewed distribution

because the mean takes the outliers into account which are lower and deflate the mean