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Chapter 4
Numerical Methods for Describing Data
Parameter -
• Fixed value about a population• Typical unknown
Suppose we want to know the MEAN length of all the fish in Lake Lewisville . . .Is this a value that is known?Can we find it out?
At any given point in time, how
many values are there for the
mean length of fish in the lake?
Statistic -• Value calculatedcalculated from a sampleSuppose we want to know the MEAN length of all the fish in Lake Lewisville.
What can we do to estimate this unknown parameter?
Measures of Central Tendency• Mode – the observation that occurs
the most often– Can be more than one mode
– If all values occur only once – there is no mode
– Not used as often as mean & median
Measures of Central TendencyMedian - the middle value of the data;
it divides the observations in half
To find: list the observations in numerical order
even is if values middle two the of average
odd is is value middle single median sample
n
n
Where n = sample size
Suppose we catch a sample of 5 fish from the lake. The lengths of the fish (in inches) are listed below. Find the median length of fish.
33 4 4 5 5 8 8 10 10
The numbers are in order & n is odd – so find the
middle observation.
The median length of fish is 5 inches.
Suppose we caught a sample of 6 fish from the lake. The median length is …
33 4 4 5 5 6 6 8 8 10 10
The numbers are in order & n is even – so find the middle two observations.
The median length is 5.5 inches.
Now, average these two values.
5.5
Measures of Central TendencyMean is the arithmetic average.
– Use to represent a population mean
– Use x to represent a sample mean
n
xx
Formula:
is the capital Greek letter sigma – it means to sum the values that
follow
parameter
statistic
is the lower case Greek letter mu
Suppose we caught a sample of 6 fish from the lake. Find the mean length of the fish.
33 4 4 5 5 6 6 8 8 10 10
To find the mean length of fish - add the observations and
divide by n.
61086543
6x
x (x - x)
3
4
5
6
8
10
Sum
What is the sum of the deviations from the mean?
Now find how each observation deviates from the mean.
0
Will this sum always equal zero?
YES
This is the deviation from the mean.
3-6-3
-2
-1
0
2
4
Find the rest of the deviations from the mean
The mean is considered the balance point of the distribution because it “balances” the positive and negative deviations.
Imagine a ruler with pennies placed at 3”, 4”, 5”, 6”, 8” and 10”.
To balance the ruler on your finger, you would need to place your finger at the mean of 6.
The mean is the balance point of a distribution
What happens to the median & mean if the length of 10 inches was 15 inches?
33 4 4 5 5 6 6 8 8 15 15
The median is . . .
5.5
The mean is . . .
61586543
6.833
What happened?
What happens to the median & mean if the 15 inches was 20?
33 4 4 5 5 6 6 8 8 20 20
The median is . . .
5.5
The mean is . . .
62086542
7.667
What happened?
Statistics that are not affected by extreme values are said to be resistant.
Is the median resistant?
Is the mean resistant? NO
YES
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =Median =
3 5 6 10 6 7 7 8 4 5 6 4 7 5 9 9 8 7 6 8
6.5
Calculate the mean and median.
6.5
Look at the placement of the mean and median in this symmetrical distribution.
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width 1.)
Mean =Median =
5.56.8
Calculate the mean and median.Look at the placement of the
mean and median in this skewed distribution.
3 5 6 10 15 7 3 3 4 5 6 4 12 5 3 4 8 13 11 9
Suppose we caught a sample of 20 fish with the following lengths. Create a histogram for the lengths of fish. (Use a class width of 1.)
Mean =Median =8.5
7.75
Calculate the mean and median.Look at the placement of the
mean and median in this skewed distribution.
3 5 6 10 10 7 10 8 9 5 6 4 9 10 9 9 10 7 10 8
Recap:• In a symmetrical distribution, the
mean and median are equal.• In a skewed distribution, the mean is
pulled in the direction of the skewness.
• In a symmetrical distribution, you should report the mean!
• In a skewed distribution, the median should be reported as the measure of center!
Trimmed mean:Purpose is to remove outliers from a
data set
To calculate a trimmed mean:• Multiply the percent to trim by n• Truncate that many observations
from BOTH ends of the distribution (when listed in order)
• Calculate the mean with the shortened data set
Mean = 23.8
Find the mean of the following set of data.
12 14 19 20 22 24 25 26 26 50
10%(10) = 1
So remove one observation from each side!
228
2626252422201914
Tx
Find a 10% trimmed.
60% of the sample was satisfied with their cell phone service.
6.0159ˆ p
What values are used to describe categorical data?Suppose that each person in a sample of 15 cell phone users is asked if he or she is satisfied with the cell phone service.
Here are the responses:Y N Y Y Y N N Y Y N Y Y Y N N
What would be the possible responses?
Find the sample proportion of the people who answered “yes”:
nsuccesses of numberˆ p
Pronounced p-hatThe population proportion is
denoted by the letter p.
Why is the study of variability Why is the study of variability important?important?• There is variability in virtually everything
• Allows us to distinguish between usual & unusual values
• Reporting only a measure of center doesn’t provide a complete picture of the distribution.
Does this can of soda contain exactly 12 ounces?
20 30 40 50 60 70
20 30 40 50 60 70
20 30 40 50 60 70
What is the mean and median of these three graphs?
A
B
C
Measures of VariabilityThe simplest numeric measure of variability is range.
Range = largest observation – smallest
observation20 30 40 50 60 70
20 30 40 50 60 70
20 30 40 50 60 70
The first two data sets have a range of 50 (70-20) but the third data set has a much smaller range of 10.
What is the range of these data sets?
A
B
C
Measures of VariabilityHow would a dotplot look if the average deviation was 0?
What does it mean to have an average deviation of 0?
1 2 3 4 5
Measures of VariabilityAnother measure of the variability in a data set uses the deviations from the mean (x – x).
20 30 40 50 60 70A
What is the mean of this distribution?
45
What is a deviation from
the mean?
1
22
n
xxs
Measures of VariabilityAnother measure of the variability in a data set uses the deviations from the mean (x – x).
Remember the sample of 6 fish that we caught from the lake . . .They were the following lengths:
3”, 4”, 5”, 6”, 8”, 10”The mean length was 6 inches. Recall that we calculated the deviations from the mean. What was the sum of these deviations?
Can we find an average deviation?
What can we do to the deviations so that we
could find an average?
The estimated average of the deviations squared is called the variance.
Degree of freedom
(explained later)
Population variance is
denoted by 2 and divided by n.
x (x - x)
3 -3
4 -2
5 -1
6 0
8 2
10 4
Sum 0
What is the sum of the deviations squared?
Remember the sample of 6 fish that we caught from the lake . . .Find the variance of the length of fish.
Divide this by 5.
First square the deviations
Finding the average of the deviations would
always equal 0!
9
4
1
0
4
1634 s2 = 6.5
(x - x)2
What could we do so that we would be able to find an average deviation?
Measures of VariabilityThe square root of variance is called standard deviation.
A typical deviation from the mean is the standard deviation.
s2 = 6.8 inches2 so s = 2.608 inches
The fish in our sample deviate from the mean of 6 by an average of 2.608 inches.
Calculation of standard deviation of a sample
1
2
n
xxs
Population standard deviation is denoted by (where n is used in the denominator).
The most commonly used measures of center and
variability are the mean and standard deviation, respectively.
Degrees of Freedom (df)
• The number of independent observations that are free to vary
Suppose we consider the sample of 6 fish where the mean is 6 inches.
Five of these values are free to be any possible length of fish!
However, once these five values occur, then the sixth value is no longer free to
vary. It MUST be a specific value in order for the deviations from the mean
(of 6) to have a sum of zero.
Thus, out of a sample of n, n - 1 observations are free to
vary.
Measures of VariabilityInterquartile range (IQR) is the range of the middle half of the data.
Lower quartile (Q1) is the median of the lower half of the dataUpper quartile (Q3) is the median of the upper half of the data
IQR = Q3 – Q1
The Chronicle of Higher Education (2009-2010 issue) published the accompanying data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
21 27 26 19 30 35 35 26 47 26 27 30 24 29 22 24 29 20 20 27 35 38 25 31 19 24 27 27 23 34 25 32 26 24 22 28 26 30 23 25 22 25 29 33 34 30 17 25 23 34 26
Find the interquartile range for this set of data.
21 27 26 19 30 35 35 26 47 26 27 30 24 29 22 24 29 20 20 27 35 38 25 31 19 24 27 27 23 34 25 32 26 24 22 28 26 30 23 25 22 25 29 33 34 30 17 25 23 34 26
First put the data in order & find the median.
17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47
26
Find the lower quartile (Q1) by finding the median of the lower half.
24
Find the upper quartile (Q3) by finding the median of the upper half.
30
IQR = 30 – 24 = 6
Which measure(s) of variability (spread) is/are resistant?
Only the IQR!
Wolf Stat Company Activity
How does the mean and standard deviation change with linear transformations?
Linear transformation rule
• When adding a constant to a random variable, the mean changes but not the standard deviation.
• When multiplying a constant to a random variable, the mean and the standard deviation changes.
An appliance repair shop charges a $30 service call to go to a home for a repair. It also charges $25 per hour for labor. From past history, the average length of repairs is 1 hour 15 minutes (1.25 hours) with standard deviation of 20 minutes (1/3 hour). Including the charge for the service call, what is the mean and standard deviation for the charges for labor?
25.61$)25.1(2530
33.8$31
25
Stat Land Game Activity ?
Move 1
How do you combine the mean and standard deviation of two independent random variables?
Rules for Combining two variables
• To find the mean for the sum (or difference), add (or subtract) the two means
• To find the standard deviation of the sum (or differences), ALWAYS add the variances, then take the square root.
baba
baba
22baba
If variables are independent
Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the times for each setup phase are independent with the following means & standard deviations (in minutes). What are the mean and standard deviation for the total bicycle setup times?Phase Mean SD
Unpacking
3.5 0.7
Assembly 21.8 2.4
Tuning 12.3 2.7minutes6.373.128.215.3 T
minutes680.37.24.27.0 222 T
Another graph- Boxplots
What are some advantages of boxplots?
• Ease of construction• Convenient handling of outliers• Construction is not subjective (like
histograms)• Used with medium or large size data
sets (n > 10)• Useful for comparative displays
BoxplotsWhen to Use Univariate numerical data
How to construct a Skeleton Boxplot– Calculate the five number summary– Draw a horizontal (or vertical) scale– Construct a rectangular box from the lower
quartile (Q1) to the upper quartile (Q3)– Draw lines from the lower quartile to the
smallest observation and from the upper quartile to the largest observation
To describe – comment on the center, spread, and shape of
the distribution and if there is any unusual features
Use for moderate to
large data sets. Don’t use with data sets of n <
10.
The five-number summary is the minimum value, first quartile, median, third
quartile, and maximum value
Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47
10 20 30 40 50
Percentages
First draw a scaleDraw a box from Q1 to Q3
Draw a line for the median
Draw lines for the whiskers
Modified boxplotsTo display outliers:• Identify mild & extreme outliers
An observation is an outliers if it is more than 1.5(iqr) away from the nearest quartile.
An outlier is extreme if it is more than 3(iqr) away from the nearest quartile.
• whiskers extend to largest (or smallest) data observation that is not an outlier
iqrQiqrQ 5.1and5.1 31
iqrQiqrQ 3and3 31
Modified boxplots are generally preferred because they provide more
information about the data distribution.
Remember the data on the percentage of the population with a bachelor’s or higher degree in 2007 for each of the 50 states and the District of Columbia.
17 19 19 20 20 21 22 22 22 23 23 23 24 24 24 24 25 25 25 25 25 26 26 26 26 26 26 27 27 27 27 27 28 29 29 29 30 30 30 30 31 32 33 34 34 34 35 35 35 38 47
10 20 30 40 50
Percentages
First, draw the scale, box and the line for
the median
Draw lines for the whiskers
Next calculate the fences for outliers.24-1.5(6) =
15
30+1.5(6) = 39
30+3(6) = 48
There is one outlier at the upper end at the distribution, but none at the lower end. Is
it extreme?
Place a solid dot for the outlier
To describe:The distribution of percent of the population with a bachelor’s degree or higher for the U.S. states and District of Columbia is positively skewed with an outlier at 47%. The median percentage is at 26% with a range of 30%.
Symmetrical boxplots Approximately symmetrical boxplot
Skewed boxplot
Notice that all 3 boxplots are identical,
but their corresponding
histograms are very different. Can you
determine the number of modes from a
boxplot?
Notice that the range of the lower half and
the range of the upper half of this distribution are
approximately equal so we can say that it
is approximately symmetrical.However, the range of
the two halves of this distribution are
definitely different sizes, so it would be
skewed in the direction of the longest side.
The 2009-2010 salaries of NBA players published on the web site hoopshype.com were used to construct the comparative boxplot of salary data for five teams.
Discuss the similarities
and differences.
Normal Curve
• Bell-shaped, symmetrical, unimodal curve
• Transition points between cupping upward and downward occur at ±
• As the standard deviation increases, the curve flattens and spreads
• As the standard deviation decreases, the curve gets taller and thinner
Let’s use our calculator to graph some normal curves
Put the following into your calculator:(Window: x: [0,20] & y: [0,0.3])
Y1: normalpdf(X,10,2)Y2: normalpdf(X,10,1.5)Y3: normalpdf(X,10,3)
What happens?
Input the following command into a graphing calculator in order to graph a normal curve with a mean of 20 and standard deviation of 3.
Y1 = normalpdf(X,20,3)(Window x: [10,30] y: [0,0.2])
Use the command 2nd trace, 7 to find the area under the curve for the: (Round to 3 decimal places.)
Lower limit: 17 Upper limit: 23 Area: ________Lower limit: 14 Upper limit: 26 Area: ________Lower limit: 11 Upper limit: 29 Area: ________
What’s my area?
Graph a normal curve with a mean of 50 and standard deviation of 5.
Y1 = normalpdf(X,50,5) (x: [30,70] y: [0,0.1])
Find the area under the curve for the following:
Lower limit: 45 Upper limit: 55 Area: ________Lower limit: 40 Upper limit: 60 Area: ________Lower limit: 35 Upper limit: 65 Area: ________
What’s my area?
What pattern do you notice?
Interpreting Center & VariabilityEmpirical Rule-
• Approximately 68% of the observations are within 1 standard deviation of the mean
• Approximately 95% of the observations are within 2 standard deviation of the mean
• Approximately 99.7% of the observations are within 3 standard deviation of the mean
Can ONLY be used with distributions that are mound shaped!
68%95%99.7%
The height of male students at PWSH is approximately normally distributed with a mean of 71 inches and standard deviation of 2.5 inches.
a)What percent of the male students are shorter than 66 inches?
b) Taller than 73.5 inches?
c) Between 66 & 73.5 inches?
About 2.5%
About 16%
About 81.5%
Measures of Relative StandingZ-score
A z-score tells us how many standard deviations the value is from the mean.
deviation standard
mean-valuescore-z
One example of standardized score.
What do these z-scores mean?
-2.3
1.8
-4.3
2.3 standard deviations below the mean
1.8 standard deviations above the mean
4.3 standard deviations below the mean
Sally is taking two different math achievement tests with different means and standard deviations. The mean score on test A was 56 with a standard deviation of 3.5, while the mean score on test B was 65 with a standard deviation of 2.8. Sally scored a 62 on test A and a 69 on test B. On which test did Sally score the best?
714.15.3
5662
z
She did better on test A.
Z-score on test A Z-score on test B
429.18.2
6569
z
Measures of Relative StandingPercentiles
A percentile is a value in the data set where r percent of the observations fall AT or BELOW that value
In addition to weight and length, head circumference is another measure of health in newborn babies. The National Center for Health Statistics reports the following summary values for head circumference (in cm) at birth for boys.
Head circumference (cm)
32.2 33.2 34.5 35.8 37.0 38.2 38.6
Percentile 5 10 25 50 75 90 95
What percent of newborn boys had head circumferences greater than 37.0 cm?
10% of newborn babies have head circumferences bigger than what value?
25%
38.2 cm