ch 5 Central Tendency and Variability

Definition Central tendency

a statistical measure that determines a single value that accurately describes the center of the distribution and represents the entire distribution of scores.

Variability

-A measure of variability usually accompanies a measure of central tendency as basic descriptive statistics for a set of scores. -The goal for variability is to obtain a measure of how spread out the scores are in a distribution -Do most people score similarly, or is there a very large difference is scores. -most of the questions we have in research are about variability, why people are not the same

Calculating the Mean

-Add up all scores in the sample. -Divide sum of all scores by the total number of scores. 𝑀=Σ𝑋/𝑁

Central Tendency and Variability

-Central tendency describes the central point of the distribution, and variability describes how the scores are scattered around that central point. -Together, central tendency and variability are the two primary values that are used to describe a distribution of scores.

Parameter-The Mean Symbol

-Used for population -Symbol:𝜇 -Pronounced "Mew"

Median Definition

the middle score in a distribution; half the scores are above it and half are below it

The Median

-If the scores in a distribution are listed in order from smallest to largest, the median is defined as the midpoint of the list (the middle score). -The median divides the scores so that 50% of the scores in the distribution have values that are equal to or less than the median.

The Mode

-In a frequency distribution graph, the mode is the category or score corresponding to the peak or high point of the distribution. -The mode can be determined for data measured on any scale of measurement: nominal, ordinal, interval, or ratio. -The primary value of the mode is that it is the only measure of central tendency that can be used for data measured on a nominal scale

Calculating the Median

-Line up the scores in ascending order. -Find the middle number. -For an odd number of scores, just find the middle value. -For an even number of scores, divide number of scores by two. Take the average of the scores around this position.

The Median (cont'd.)

-One advantage of the median is that it is relatively unaffected by extreme scores. -Thus, the median tends to stay in the "center" of the distribution even when there are a few extreme scores or when the distribution is very skewed. In these situations, the median serves as a good alternative to the mean.

Measures of Variability

-Range :From the lowest to the highest score -Variance: Average square deviation from the mean -Standard Deviation: Variation from the sample mean

The Mean

-The mean is the most commonly used measure of central tendency. Computation of the mean requires scores that are numerical values measured on an interval or ratio scale. -The mean is obtained by computing the sum, or total, for the entire set of scores, then dividing this sum by the number of scores.

The Mean in Symbols

-The mean of a sample is an example of a statistic, whereas the mean of a population is an example of a parameter. -The symbols we use depend on whether we are referring to the mean of a sample or of a population

The Range

-The range is the total distance covered by the distribution, from the highest score to the lowest score. -Simplest measure of variability, least useful

Statistic-The Mean Symbol

-Used for Sample -Symbol:𝑀 𝑜𝑟 𝑋 ̅ -Pronounced "M" or "X bar"

Statistical Symbols

A descriptive value for a population is called a parameter and a descriptive value for a sample is called a statistic.

Is it possible to compare 2 or more sets of data?

It is possible to compare two (or more) sets of data by simply comparing the average score (central tendency) for one set versus the average score for another set.

Multimodal distribution

More than two modes, or most common scores

Mode definition

Most common score

Unimodal distribution

One mode, or most common score

Standard Deviation Symbol

Population: curvy o Sample: s

Variance Symbol

Population: curvy o^2 Sample: s^2

The Standard Deviation

Standard deviation measures the standard (average) deviation (distance) between a score and the mean. Standard Deviation (SD) Average deviation of scores from the mean A smaller SD indicates less variability The calculation of standard deviation can be summarized as a four-step process:

Examples of Range

Test scores: 23 to 98 or(98-23 = 75) Incomes: $15,300 to $76,500 or ($76,500-15,300 = $61,200) Temperature: 62° to 85° or (85°- 62° = 23°) Highest score - lowest score Or Min to Max Only considers extreme values -- not very useful

Bimodal distribution

Two modes, or most common scores

The mode is used best for..

data are group/nominal

What type of statistic is central tendency?

descriptive statistic because it allows researchers to describe a set of data in a very simplified, concise form.

Range equation

range=𝑋_ℎ𝑖𝑔ℎ𝑒𝑠𝑡−𝑋_𝑙𝑜𝑤𝑒𝑠𝑡=10−1=9

The median is used best for..

score data that falls in a skewed distribution or one with one or more outliers.

Mean is used best for..

score data that falls in a symmetric distributions.

Mean Definition

the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

Central tendency

three slightly different ways to describe what is happening in the center of a distribution of data -Mean -Median -Mode