P. 377Lesson 52 – Measures of Center and Spread

P. 377-378

What is the Center? A central value within the data set.What is it used for? To describe and compare data sets.What does it show? A typical value for the data set.How do you measure it? There are 3 Measures Of Center (aka Central Tendency):

P. 378

Measures

Of CenterSymbol(s) Definition How do you find it?

Must you order

the data set

to find it?

1 MeanΜ

x̄

Average of the

data values

Add up all the values, then divide by the

number of values.No

2

Median

aka 50th Percentileaka 2nd Quartile

aka 𝑄2

MdMiddle value

If there are an odd number of values,

the median is the middle one.

If there are an even number of values,

the median is the average of the 2

middle ones.

Yes

3 Mode MoMost frequently

occurring valueCount the frequency for each value. Recommended

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And the Mode? ______

And the Mode? ______

If the mean increases after a single data point is added to a set of data, what can you tell about this data point?

That the data point added was larger than the mean of the set.

P. 379

And the Mode? ______

And the Mode? ______

P. 379

What is the Spread? How close each data value is to the center of the data.What is it used for? To describe how consistent the data values are.What does it show? 1) The distance between the data values and the center.

2) The distance between the data values and each other.How do you measure it? There are 3 Measures Of Spread.

Summarizing the dataset can help us understand the data, especially when the dataset is large. The Measures of Center that we mentioned earlier summarizes the data into a single value that is typical or representative of all the values in the dataset, but this is only part of the picture. Measures of Spread summarize the data in a way that shows how scattered the values are, and how much they differ from the mean value.

The 3 Measures Of Spread:

Measures

Of SpreadSymbol Definition How do you find it?

Measures the

spread of

the…

Must you

order

the data set

to find it?

1 Range --Difference between the

greatest and least data valuesGreatest minus least

Entire data

setNo

2Interquartile

RangeIQR

Difference between the 3rd

and 1st quartiles𝑄3 minus 𝑄1

Middle half

of the dataYes

3Standard

Deviation

σ

std(X)

σX

s

Average of the distance

between individual data

values and the mean

Where 𝑥1 is the 1st data value,𝑥2 is the 2nd data value, etc.,n is the number of data values,and x̄ is the mean

Data

relative

to the

mean

No

P. 379-381

(There’s actually a 4th MOS called Variance, but we won’t be covering it here.)

Term aka aka Definition

1 Quartiles -- -- Values that divide a data set into 4 equal parts

2 2nd Quartile 𝑸𝟐 50th Percentile The median of the whole data set

3 1st Quartile 𝑸𝟏25th Percentile

Lower QuartileThe median of the lower half of the data set

4 3rd Quartile 𝑸𝟑75th Percentile

Upper QuartileThe median of the upper half of the data set

5 Interquartile Range IQR -- The difference between the 3rd and 1st quartiles (𝑸𝟑 − 𝑸𝟏)

P. 379The Range is the easiest to calculate: Subtract the lowest valuerom the highest value. (Can it ever be negative?)

Next comes the Interquartile Range, or IQR.

Here are some definitions:

Here’s the sequence:

1) Order the data (if necessary)2) Find the median (𝑸𝟐)3) Find the 1st quartile (𝑸𝟏)4) Find the 3rd quartile (𝑸𝟑)5) Calculate the IQR (𝑸𝟑 minus 𝑸𝟏)

P. 379

Midrange = 𝑳𝒐𝒘𝒆𝒔𝒕+𝑯𝒊𝒈𝒉𝒆𝒔𝒕

𝟐=

77 + 93

2=

170

2= 85

Midquartile = 𝑸𝟏+𝑸𝟑

𝟐=

80.5 + 91.5

2=

172

2= 86

Two new terms

P. 380

Midrange = 𝑳𝒐𝒘𝒆𝒔𝒕+𝑯𝒊𝒈𝒉𝒆𝒔𝒕

𝟐=

2 +14

2=

16

2= 8

Midquartile = 𝑸𝟏+𝑸𝟑

𝟐=

4 + 10

2=

14

2= 7

P. 380-381

The Standard Deviation is used to tell how close the data points are to the mean. A low standard deviation value means the data points tend to be very close to the mean. A high standard deviation value means the data points are spread out.Here’s the formula for finding the SD of the data set {𝑥1, 𝑥2, … 𝑥𝑛}, with n elements and mean x̄.

Plus the Midrange and Midquartile.

Mr. Morris gave his algebra class a test, the results of which are listed below.

68, 92, 74, 75, 86, 90, 92, 81, 60, 82, 77, 80

Shania was absent on the day of the test and had to take the test late. She earned a score of 99. Which measure of the class's test results did Shania's score most change?

A. IQRB. MeanC. MedianD. Range