Measures of

Central Tendenc

y

3.1

●Analyzing populations versus analyzing samples●For populations

We know all of the data Descriptive measures of populations are called

parameters Parameters are often written using Greek letters

( μ )

●For samples We know only part of the entire data Descriptive measures of samples are called statistics Statistics are often written using Roman letters ( )

PARAMETER VS. STATISTIC

x

●The arithmetic mean of a variable is often what people mean by the “average” … add up all the values and divide by how many there are●Compute the arithmetic mean of6, 1, 5●Add up the three numbers and divide by

3(6 + 1 + 5) / 3 = 4.0●The arithmetic mean is 4.0

ARITHMETIC MEAN

●The arithmetic mean is usually called the mean●For a population … the population mean

Is computed using all the observations in a population

Is denoted μ Is a parameter

●For a sample … the sample mean Is computed using only the observations in a sample

Is denoted Is a statistic

MEAN

x

Whole Class (Population)Height in InchesFind the arithmetic mean

5 Students (Sample…statistic)Height in InchesFind the arithmetic mean

LET’S TRY IT

●The median of a variable is the “center”●When the data is sorted in order, the

median is the middle value●The calculation of the median of a

variable is slightly different depending on If there are an odd number of points, or

The middle number If there are an even number of points

Take the 2 middle numbers and find the mean

MEDIAN

●An example with an odd number of observations (5 observations)●Compute the median of6, 1, 11, 2, 11●Sort them in order1, 2, 6, 11, 11●The middle number is 6, so the median

is 6

MEDIAN

●An example with an even number of observations (4 observations)●Compute the median of6, 1, 11, 2●Sort them in order1, 2, 6, 11●Take the mean of the two middle values(2 + 6) / 2 = 4●The median is 4

MEDIAN

●The mode of a variable is the most frequently occurring value●Find the mode of6, 1, 2, 6, 11, 7, 3●The values are1, 2, 3, 6, 6, 7, 11●The value 6 occurs twice, all the other values

occur only once●The mode is 6

MODE

●Qualitative dataValues are one of a set of categoriesCannot add or order them … the mean and median do not exist

The mode is the only one of these three measurements that exists

●Find the mode ofblue, blue, blue, red, green●The mode is “blue” because it is the

value that occurs the most often

WEIRD MODE

●Quantitative dataThe mode can be computed but sometimes it is not meaningful

Sometimes each value will only occur once (which can often happen with precise measurements)

●Find the mode of5.1, 6.6, 6.8, 9.3, 1.9●Each value occurs only once●The mode is not a meaningful measurement●Mode is what is used in elections!

MODE (NO REPEATS)

The mean and the median are often different

This difference gives us clues about the shape of the distribution Is it symmetric? Is it skewed left? Is it skewed right?Are there any extreme values?

SHAPE

Symmetric – the mean will usually be close to the median

Skewed left – the mean will usually be smaller than the median

Skewed right – the mean will usually be larger than the median

SHAPE

●If a distribution is symmetric, the data values above and below the mean will balanceThe mean will be in the “middle”The median will be in the “middle”

●Thus the mean will be close to the median, in general, for a distribution that is symmetric

SYMMETRIC

● If a distribution is skewed left, there will be some data values that are larger than the others The mean will decrease The median will not decrease as much

●Thus the mean will be smaller than the median, in general, for a distribution that is skewed left

SKEWED LEFT

● If a distribution is skewed right, there will be some data values that are larger than the others The mean will increase The median will not increase as much

●Thus the mean will be larger than the median, in general, for a distribution that is skewed right

SKEWED RIGHT

Birth Weights

5.8 7.4 9.2 7.0 8.5 7.6

7.9 7.8 7.9 7.7 9.0 7.1

8.7 7.2 6.1 7.2 7.1 7.2

7.9 5.9 7.0 7.8 7.2 7.5

7.3 6.4 7.4 8.2 9.1 7.3

FINDING MEAN AND MEDIAN ON CALCULATOR

•Find the mean and median•Make a Histogram to discuss the shape of the data•How to sort data in lists

Mean The center of gravity Useful for roughly symmetric quantitative data

Median Splits the data into halves Useful for highly skewed quantitative data

Mode The most frequent value Useful for qualitative data

SUMMARY (IMPORTANT STUFF!)