Download Lesson 2 Measures Center

Unit 7: Descriptive Statistics
Lesson 2: Summarizing Data
Objectives:
1. To use a graphing calculator to find measures of central tendency.
2. To determine which measure of central tendency is appropriate for a data set.
Measures of central tendency
In addition to graphing a data set, a measure of its center or average also helps to
describe the data. The two most common measures of center are the mean and median.
Mean:
sum of data values
number of data values
x + x + ... + xn
= 1 2
n
x=
Raw Data
n
=
åx
i=1
i
n
sum (each data value ´ frequency)
x=
number of data values
Frequency
Table with
Individual Data
Values
n
=
i=1
i
i
n
n
=
x=
Frequency
Table
with Grouped
Data
å x × f (x )
å xi × f (xi )
i=1
åf
xi = data value
f (xi ) = frequency of xi
mi = midpoint of interval
å is short for "add them all up"
sum (each midpoint ´ frequency)
number of data values
n
=
å m × f (x )
i=1
i
i
n
n
å m × f (x )
=
åf
i
i
i=1
Median: Firstly arrange the data in order. The median is
bottom or top.
n +1
observations from the
2
If there is an odd number of measurements, the median = middle value.
If there is an even number of measurements, the median = average of the two middle
values.
Mode: The most common data value. This is the only measure of central tendency that
can be found for qualitative data. However, it is rarely used for quantitative data.
Doing these calculations by hand can be time consuming. Thankfully the TI-84 calculator
can do these for you, and in this unit you may always use a calculator to find any
summary statistics.
How to find summary statistics on the TI-84 calculator

Enter your data into a list. There are two ways to do this: using a single list, or using
a data list and a frequency list.
Option 1 (L1 contains data values)

Option 2 (L2 contains frequencies)
To calculate the summary statistics, press STAT, scroll over to CALC, and choose
option 1:1-Var Stats
Option 1
Option 2
List:L1
FreqList: (leave blank)
List:L1
FreqList:L2
Interpreting the output for one variable statistics*
Symbol Meaning
mean
x
x
sum of all data values
squares each data value, then
x2
adds them up
sx
sample standard deviation
x
population standard deviation
n
MinX
Q1
Med
Q3
MaxX
number of data values
minimum (smallest) data value
value of first (lower) quartile
median
value of third (upper) quartile
maximum (largest) data value
Note
Don’t use this standard deviation (you will
learn about this in AP Statistics)
In Algebra 2 Trig we always use this one for
the standard deviation.
These five values are known collectively as
the five-number summary.
* We will learn about most of these statistics over the next few classes.
1.
Find the mean and median of the number of siblings each student has (Example 2
from Lesson 1 notes).
2.
Find the mean and median of the hand widths you measured last class (Example 3
from Lesson 1 notes).
3.
Give an example of a set of five positive numbers whose median is 10 and whose
mean is larger than 10.
4.
In a small town of 1000 residents everyone earns $80,000 per year.
a. What are the median and mean incomes for the town?
5.
b.
Donald Trump likes the town as it reminds him of his childhood, so he decides
to move in. He earns $1,000,000,000 per year. What are the new median and
mean incomes?
c.
Describe how Donald Trump’s earnings impact the mean and median incomes.
The brightness of celestial bodies depends on many factors, two of the most
important being the distance from Earth and size. The eight brightest objects in the
night sky are listed below with their approximate distance from Earth (in light
years).
Celestial Body
Distance from Earth (light years)
Moon
0.000000038
Venus
0.0000048
Jupiter
0.000067
Mars
0.0000076
Mercury
0.0000095
Syrius
8.6
Canopus
310
Saturn
0.00014
a.
Calculate the mean and median for these distances.
b.
Would the typical distance of these celestial bodies best be communicated using
the mean or the median? Why?
6.
Suppose that a sample of 100 homes in the metropolitan Phoenix area had a median
sales price of $300,000. The mean value of these homes was $1,000,000. Explain
how this could happen. Why might the median price be more informative than the
mean price in describing a typical house price?
7.
Suppose the mean annual income for a sample of one hundred Minneapolis
residents was $50,000. Do you think the median income for this sample would have
been greater than, equal to, or less than $50,000? Explain.
In summary, the three measures that describe the middle of a data set are the ___________,
_______________, and ___________. If a data set contains unusually large or small values, the
most appropriate measure to describe a “typical” value is the ________________.
Homework: (to be completed in your math workbook)
1.
Using the “hours of sleep” data set from the Algebra 2/Trig survey results:
a. Draw a histogram of the data set.
b. Find the mean and median of the data set.
c. Explain whether the median or mean best describes a typical value in the
dataset.
2.
Find an interesting data set online containing at least 30 values.
a. Graph your data set using an appropriate graph.
b. Find the mean and median of the data set.
c. Explain whether the median or mean best describes a typical value in the
dataset.