Chapter 5Estela Molina and Neil Charles

Finding the center:The Median

Median- the middle values that divides the histogram into two equal areas.

For a unimodal, symmetric distribution, it’s easy to find the center—it’s just the center of symmetry.Formula: (n+1)/2

Spread

When we describe a distribution numerically we always report a measure of its spread along with its center.Range: of the data is defined as the difference between the maximum and minimum values. Range= Max- Min

Spread: The interquartile Range

The difference between the quartiles tells us how much territory the middle half (50%) of the data covers and is called the Interquartile range (IQR).

IQR = upper quartile – lower quartileThe lower and upper quartiles are also known as the 25th and 75th percentiles of the data.

5 Number Summary

Reports its median, quartiles, and extremes (maximum and minimum).Example: The five-number summary for the ages at death for rock concertgoers who died from being crushed is

5 Number Summary: Steps

The 5 Number summary can be graphed in a Boxplot.

1. Draw the vertical axis. graph the median, quartile 1,and quartile 3.

2. Graph upper fence and lower fence. Upper fence = Q3 +1.5 (IQR) Lower fence= Q1+1.5(IQR)

3. Graph the maximum and minimum.

4. Graph outliers

How to find the mean

The means the point at which the histogram would balance.

Standard deviation

How far each value is from the mean.★ Standard deviation and mean are appropriate only for

symmetric data.

Question 31A survey of major universities asked what percentage of incoming freshmen usually graduate “on time” in four years. Use the summary statistics given to answer these questions.

A. Would you describe this distribution as symmetric or skewed? Explain.B. Are there any outliers? Explain.C. create a boxplot of these data.D. write a few sentences about the graduation rates.

% on time

count 48

Mean 68.35

Median 69.90

Std. Dev 10.20

Min 43.20

Max 87.40

Range 44.20

25th %tile 59.415

75th%tile 74.75

Question 31A. Slightly left skewed.B. There are no outliers in this graph. Because, all of the information given in the 5 number summary was with in

the upper and lower fence.

C. The 48 universities graduate, on average, about 68% of the freshman on time with percents ranging from 43% to 87%. The middle 50% lies between 59% and 75% who graduate in 4 years.

Question 35

How fast do horses run? Kentucky Derby Winners top 30 miles per hour, as shown in the graph on the next page. In fact, this graph shows the percentage of derby winners that have run slower than a given speed. Note that a few have won running less than 33 miles per hour, but about 95% of the winning horses have less than 37 miles per hour.

A. Estimate the median winning speed.B. Estimate the quartiles.C. Estimate the range and the IQR.D. Create a boxplot of these speeds.E. Write a few sentences about the speeds of the kentucky Derby winners.

Answer 35

A. To find the median, find the number of the graph which lines up with 50%. So the answer is 36 miles per hour.B. To find Q1, find the number that lines up with 25%, which is 34.5mph and to find Q3, find the number that lines

up with 75% on the graph, which is 36.5 mph.C. The range is 7 mph from 31-38 mph. The IQR= Q3-Q1= 2 mph.

D. The median winning speed is 36 mph with a max of 38 and min of 31 mph. The middle 50% are between 34.5 and 36.5 mph with an IQR of 2 mph. About 5% of the derby winners raced at speeds more than 37 mph.