StatisticsChapter 2 Exploring Distributions

Section 2.3Measures of Center and Spread

Measures of Center

Mean (average): Sum of x’s / total number of values It is the balance point of a distribution

Median is the value the divides the data into 2 halves. The middle number or average of middle 2

numbers. 1,3,6,8,9 Median is 6 1,3,7,9,10,15 Median is (7+9)/2=8

𝑥ҧ= σ𝑥𝑛

Effect of Outliers

An extreme outlier has a larger effect on the mean than the median. Why? 1,4,6,8 Find the mean and median 1,4,6,8 and 55: Now find the mean and

median Remember the mean is the balance

point of a distribution and the median is the middle number.

Median and the Spread

Use the Speed of Mammals to investigate median and the 5-Number Summary:

Find the median of the Predators:

Find the lower or first quartile of the Predators: (omit the median)

Find the upper of third quartile of the Predators: (omit the median)

What are the minimum and maximum values for Predators:

The 5-Number Summary and gives a measure of center and spread.

Now do the same for the non-predators:

Speed (mi/h) Predator (1=Y, 0=N30 130 170 130 039 140 025 045 042 132 020 048 040 050 111 035 012 040 0

5-Number Summary and Box and Whisker Plot

Use the 5-Number Summary to create a box and whisker plot.

When comparing two groups use the same scale and place box and whisker plots on top of each other.

Use the mammal speed data to compare wild mammals to non-wild mammals with box and whisker plots.

Practice and Discussion Questions

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Page 69 P16, 18, 19, 21

IQR: Interquartile Range The IQR is a measure of the spread of

the middle 50% of the data.

IQR=Q3-Q1

This can be use to determine if a value is an outlier. If the value is more than 1.5 times the IQR

from the nearest quartile. Outliers in a Boxplot.

Measure Spread around the Mean

Measuring Hand Span Activity Using a ruler, measure your hand span to

the nearest mm. Write your result on the board. Enter all data into calculator. Find the 5 Number Summary of the data. ▪ (1-var stats)▪ Create a box plot

Enter L1-mean into L2 Use 1-var stats to find the sum of this: x

Meaning of the sum of differences

Think about what subtracting the mean (x) from your data values does. It tells how far your hand span is from the

class average. ▪ + and your hand span is larger▪ - and your hand span is smaller

Now think about what summing all of your differences does. What was the result? Differences from the mean will always sum to be

Zero.

“Average” difference from mean Rapid Fire Example , Rapid Fire 2 A measure of the average difference from the

mean gives you a kind of level of consistency. Think of a target in an archery contest.

If you wanted to measure someone’s accuracy, how would you suggest doing it?

Come on…Just hit it!!

Standard Deviation (SD)

The standard deviation of a data set is typically used when the data is roughly normally distributed. It measures the “average” distance from the mean. This is a measure of spread. (how far are the arrows

spread apart) To calculate a SD, we must figure out a way to

find the average difference from the mean, but we know that the sum of the differences is always = zero. So we square the difference, then sum that, then divide

to find average sum of squares, then take the square root.

1

)( 2

n

xxs

Variance

The SD squared: (s2) is called the variance.

It is not used as often, but is important to know what it is, in case it is a piece of information given.

1-var stats on the Calculator

You can get the SD from the calculator. Sx is the SD of a sample of data. σx is the SD of an entire population (we

will discuss the difference later) Note the other sums and values you

can get. Σx will give the sum of you list of values Σx2 will give the sum of squared values n is your total number of data values in

the list Then is your 5-Number Summary

Frequency Table

Frequency Tables show the values and the frequency that they occur in the data set.

You can easily find the mean of this data:

You can find the SD:

It is easy to find with Calculator: L1:x values, L2: frequencies, 1-var stat

L1,L2

n

fxx

)(

1

)( 2

n

fxxs

Practice and Discussion Questions

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Homework

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