Download 8.3 Describing the Average and Spread of Data

8.3 Describing the Average
and Spread of Data
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Measures of Central Tendency
(Averages)
What is an average?





How the data clusters
How the data centers
A typical number
A central number
A representative number
Measures of Central Tendency
1.
2.
3.
4.
Mean
Median
Mode
Midrange
Measures of Central Tendency
(cont.)
Arithmetic Mean x




The sum of all the data values divided by
the number of data values
x  x  x  ...  x
x
n
1
2
3
n
A balance point
It is affected by extreme values
Measures of Central Tendency
(cont.)
Median

The numerical value that is the middle
number in an ordered list of the data
 If there is an even number of data points, find
the mean of the two numbers in the middle of
the ordered list

The value about which the data set is
equally split
Measures of Central Tendency
(cont.)
Mode

The numerical value(s) that occurs most
frequently
 Bimodal
 No mode


Most common data point
Unaffected by all the other scores
Measures of Central Tendency
(cont.)
Midrange


The mean of the largest and smallest
values in the set of data
The point midway between the largest and
smallest numbers in the data set
Measures of Dispersion
(Spread or Scattering)
What is a measure of dispersion?


How the data spreads out
How the data is distributed
Measures of Dispersion
1.
2.
3.
4.
Range
Interquartile Range
Variance
Standard Deviation
Measures of Dispersion (cont.)
Range

The difference between the largest and
smallest values in the data set
Measures of Dispersion (cont.)
Interquartile range IQR


The range of the middle half of the data
IQR = Q3 – Q1 where Q3 is the third
quartile point and Q1 is the first quartile
point
To find the IQR
1.Find the median of the data set. Mark its
location and call it Q2.
2.You now have two subsets of the data. Find
the median of the lower subset. Mark its
location and call it Q1.
3.Find the median of the upper subset. Mark
its location and call it Q3.
4.Subtract Q1 from Q3. This value is the IQR.
Measures of Dispersion (cont.)
Variance σ2


The mean of the squared deviations from
the mean of the data set
2
2
2
2
(
x

x
)

(
x

x
)

(
x

x
)

...

(
x

x
)
2
3
n
2  1
n
To Find the Variance
1. Find the mean, x , of the data set
2. For each number xi in the data set,
calculate the deviation of that number from
the mean, xi - x
3. Square all the deviation scores obtained in
step 2, (xi - x)2
4. Find the mean of the squared deviations
Measures of Dispersion (cont.)
Standard Deviation σ

The square root of the variance
Box and Whisker Plots
A graphical display of the median,
range, and interquartile range
To make a box and whisker plot:
1.
2.
3.
Find the range, median, and IQR of the
data
Mark an appropriate scale on either a
vertical or horizontal axis
Place points for the lowest score, Q1, Q2,
Q3, and the highest score
4.
5.
6.
7.
Draw a box whose top (or right side) is at
Q3 and whose bottom (or left side) is at
Q1
Divide the box into two portions with a
line segment at Q2
Draw a line segment from the top of the
box (or right side) to the highest score
Draw a line segment from the bottom of
the box (or left side) to the lowest score