Download Lesson 19 Sep PS - Anderson School District 5

Prob and Stats, Sep 19
Grouped Data Computations and Percentiles
Book Sections: 3.3-3.4
Essential Questions: How do I compute and use statistical values? What
is grouped data, and how can I use it in statistical computations?
What are percentiles and how are they used?
Standards: PS.SP.ID.2, .3
Grouped Data
• Grouped data – data that is summarized in a frequency
table
• Tabular only – no raw data is known
• We use what we know – class width and frequency – to
compute statistics
• Mean
• Standard deviation
• These computations are approximate values
Grouped Data Class Width
• We use the class midpoint as the value of the class in any
grouped statistical computation.
• The midpoint formula: midpo int 
(class low  class high )
2
where classlow and classhigh are the class boundries
Example
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
1
2
3
5
4
3
2
Definition
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
1
2
3
5
4
3
2
Definition: Modal
Class – The class
with the greatest
frequency
Finding the Mean of a Group
___
X
 f  Xm

n
Where f is the class frequency, Xm is
the midpoint, and n is the total
frequency.
Making it Work
Add two more columns to the table, as shown
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
1
2
3
5
4
3
2
Making it Work
___
Add two more columns to the table, as shown
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
1
2
3
5
4
3
2
X
MidPoint (Xm)
 f  Xm

n
f ·Xm
Finding the Standard Deviation of a Group
We fight the this battle via variance, like last time

2

f

X
m   f  X m  / n
2
s 
n 1
2

Where f is the class frequency, Xm is
the midpoint, and n is the total
frequency and Xm2 is the square of
the midpoint.
Making it Work
Add three more columns to the table, as shown
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
1
2
3
5
4
3
2
Making it Work
Add two more columns to the table, as shown
Class (interval)
5.5-10.5
10.5-15.5
15.5-20.5
20.5-25.5
25.5-30.5
30.5-35.5
35.5-40.5
Frequency (f)
MidPoint (Xm) f ·Xm
1
2
3
5
4
3
2

2

f

X
m   f  X m  / n
2
s 
n 1
2

f ·Xm2
Help Me Mr Texas!
• Lucky you, the calculator can do it!
Calculator needs – midpoints in L1 and frequencies in L2
Calculator Setup
• You must use two lists to accomplish this
calculator computation
1. Start with [STAT]  select {EDIT}
2. Put class midpoints into L1: (lo + hi)/2
3. Put frequencies into L2: one per midpoint
4.
Press [STAT]  select {CALC},
1-Var stats L1,L2
• Only use mean and standard deviation
Example
Compute the mean and standard deviation of this frequency
table:
Class (interval)
2.1-2.7
2.8-3.4
3.5-4.1
4.2-4.8
4.9-5.5
5.6-6.2
Frequency (f)
12
13
7
5
2
1
Big Example
Compute the mean and standard deviation of this frequency
table:
Class (interval)
0-4
5-9
10-14
15-19
20-24
25-29
Frequency (f)
140
153
207
187
168
145
Partitioning Data
• In the first half of this unit, we divided a
data set into 4 equal parts by finding
quartiles (Q values)
• Now, we are going to conceptually do a
similar set division into 100 equal parts
called percentiles.
 Reality says you need more than 100 pieces of data to
accomplish this, but we will use theory to do it on less.
Percentiles
• A percentile is a measure of position that a
piece of data falls in if the set were
divided into 100 equal parts.
• This statistic would indicate how a value
compares to the rest of the data as a
percent in ascending order.
• Percentiles are usually computed in very
large data sets.
How are Percentiles Used?
• Percentiles indicate how a single data
value compares to all others, ranked as a
percent.
• Percentiles are frequently used in
education and health-related fields to
indicate how an individual compares to a
group. Ex – SAT scores.
Interpreting Percentiles
Percentiles indicate a rank relative to the data
set, not some external measure. It is an
internal comparison.
If the weight of a six-month-old infant is at
the 78 percentile, that child weighs more than
78% of all 6-month olds, not that the infant
weighs 78% of some ideal weight.
Percentile Computations
To compute the percentile of a particular data
value, x, use the formula
Percentile rank of x =
B  0.5E
100
N
Where B is the number of data values lower
than the value of x, E is the number of terms
equal to x, and N is the total number of
values in the set.
Percentile Computations II
%ile =
B  0.5E
100
N
You will have to think your way through this
formula.
Notice that the score itself (x) is not a part of
this computation. Everything is relative to a
position in the scores.
Percentile Computations III
%ile =
B  0.5E
100
N
Breaking it down: if x is a value you want the
percentile of
• B is the number of items lower than x
 Big problem: rank – 1, Small problem: Count those less
• E is equal to, how many are equal to x? (1 is min)
• N is the total. How many are there?
• Truncate the answer (drop decimal values)
Example 1
The Big Problem
Of 32,578 scores on the September SAT math test, Yanno was the
27,416th highest, and was tied with 77 others with his score. What
is the percentile of Yanno’s score?
B  0.5E
100
N
Example 2
The Small Problem
B  0.5E
100
N
In an English 4 class, the following were the grades on senior
projects:
65, 68, 69, 71, 71, 73, 74, 75, 75, 77, 79, 79, 80, 81, 83, 83, 83,
84, 88, 88, 89, 93, 93, 95, 95, 96, 96, 98, 99
Jane made an 83. What is the percentile of her score?
Example 3
On the UN Quality of life survey, Anderson, SC was rated 13,218
highest for quality of life, and was tied with 48 other cities and
towns, world-wide. There were 27,212 cities and towns evaluated
in the report. What is the percentile of Anderson’s rank?
B  0.5E
100
N
Your Example
B  0.5E
100
N
Tyrone practiced his skills in tiddlywinks two hours a day. Sure
enough, he won his high school tiddlywinks contest, making 98 out of
100 attempts. The scores were as follows:
22, 29, 31, 35, 36, 40, 42, 45, 50, 55, 59, 61, 67, 73, 77, 80, 87, 88,
93, 97, 98
What was the percentile of Tyrone’s score score?
B=
E=
N=
Percentile Computations Backwards
%ile =
B  0.5E
100
N
If given a percentile and asked for B, E, or N,
convert the % to a decimal (p) and think of
the formula like this:
B  0.5E
p
N
Class work: Classwork 9/19/16, 1-10
Homework: HW Due 9/20/16, 1