DESCRIPTIVE STATISTICS ( Measures of Central Tendency &Measures of Position)

measure of central tendency

is a summary measure that attempts to describe a whole set of data with a single value that represents the middle or center of its distribution.

Sample means

drawn from the same population tend to vary less than other measures of center.

Mean

A disadvantage of it is that just one extreme value (outlier) can change the value of the mean substantially. (Using the following definition, we say that it is not resistant.)

Median

the middle value when the data values are arranged in order of increasing (or decreasing) magnitude.

• population median: 𝜂; sample median: 𝑥(x tilde) or 𝑀𝑒d

eta

“𝜂”

Median2

*does not change by large amounts when we include just a few extreme values,

it is a resistant measure of center.

does not directly use every data value

Mode

The value(s) that occurs with the greatest frequency.

can be found with qualitative data.

unimodal

one mode

bimodal

two modes

trimodal

three modes

multimodal

four or more modes

Midrange

the value midway between the maximum and minimum values in the original data set.

it is very sensitive to those extremes so it is not resistant.

higher mean

suggests a greater overall tendency

lower mean

Inindicates a lower typical value

Higher median

suggests a generally higher tendency,

lower mode

indicates a generally lower trend

Mode2

*Indicates the most common outcome(Jamieson, 2004).

Mean3

~Best Used When… Data is continuous and normally distributed, examples scores and average income

Mean4

$Avoid When… Example Research Areas Data is skewed or has outliers

Median3

~`Best Used When…Data is skewed, has outliers, or is ordinal, home prices, likert scales responses (avoid when data is normally distributed)

Mode3

~Best Used When…Data is categorical or has multiple peaks Measure~examples Most common product preference, popular job titles (avoid when Data is continuous without clear peaks)

measure of location or position

determines the position of a single value in relation to other values in a sample or a population data set

Quartiles

______ are measures that divide the data set into four equal parts.

Quartile 1

25% falls below the first quartile or lower quartile.

Quartile 2

50% falls below the second quartile or median.

Quartile 3

75% falls below the third quartile or upper quartile.

Deciles

__________ are measures that divide the data set into ten equal parts. There are 9 deciles

first

10% falls below the_______decile.

second

20% falls below the______decile.

third

30% falls below the_______ decile.

Median

5 th decile = _______

Percentiles

________ are measures that divide the data set into 100 equal parts. There are 99 percentiles that determine the points below which certain percentages of observations would fall.

Higher percentiles

means a value is positioned higher in the dataset.

wider IQRs

suggests greater variability in the middle 50% of the data.

Low decile: High Decile

_________means the value is among the lowest in the distribution while a_______ means the value is among the top in the distribution.

Percentile

It is commonly used in standardized testing and growth charts.

Example Applications Test scores (e.g., SAT, GRE), growth charts (e.g., a baby in the 25th percentile for weight

Quartiles (Q1, Q2, Q3)

It used When analyzing data spread and identifying outliers, example Income distribution, box plots, detecting skewnes

Decile

Used in income distribution and economic research to analyze wealth, example application Income classification (e.g., top 10% of earners), dividing test scores into performance levels