27 Jul Brilliant Answers
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Unit 6 Assignment
Using the data set entitled “Sample dataset of weight and height” found in doc sharing, calculate the mean, median, and standard deviation of EITHER the weights OR heights listed
Also provide EITHER a 5-number summary (minimum, first quartile, median, third quartile, and maximum) OR a boxplot
Lastly, based on the statistics you generated, describe the data in your own words , noting any trends that you see
Measures of Spread – Standard Deviation
STANDARD DEVIATION is the measure of spread most commonly used with the mean
It tells us the “average” (standard) amount of “variation” (deviation) for the distribution – i.e., how much variation to expect from a distribution
The formula looks and sounds very complex but it is actually quite simple when broken down
Measures of Spread – Standard Deviation
Method for calculating the standard deviation
Step 1. Calculate the mean
Step 2. Subtract the mean from each observation THEN square the difference
Step 3. Sum the squared differences (i.e., add up the values you got in Step 2)
Step 4. Divide the sum of the squared differences by n – 1 (i.e., divide the value you got in Step 3 by (n – 1))
Step 5. Take the square root (i.e., take the square root of the value from Step 4)
Measures of Spread – Standard Deviation
The standard deviation basically tells us how widely or tightly the observations are distributed
The standard deviation and mean should be used to characterize a distribution only when the data are normally distributed (think bell-curve)
Measures of Spread – Standard Deviation
Find the standard deviation for these incubation periods (in days) for hepatitis A:
27, 31, 15, 30, 22
We will go through this example step by step
Step 1. Calculate the mean
Answer on next slide
Measures of Spread – Standard Deviation
Mean = 25
(27 + 31 + 15 + 30 + 22) = 125
125 / 5 = 25
Step 2. Subtract the mean from each observation THEN square the difference
Answer on next slide
Measures of Spread – Standard Deviation
Subtract mean from each observation
27 – 25 = 2
31 – 25 = 6
15 – 25 = -10
30 – 25 = 5
22 – 25 = -3
Square the differences:
22 = 4
62 = 36
-102 = 100
52 = 25
-32 = 9
Step 3. Sum the squared differences
Answer on next slide
Measures of Spread – Standard Deviation
4 + 36 + 100 + 25 + 9 = 174
Step 4. Divide the sum of squared differences by (n -1). This will yield the VARIANCE.
Answer on next slide
Measures of Spread – Standard Deviation
Variance = 174 / (n – 1)
174 / (5 – 1) =
174 / 4 = 43.5
Variance = 43.5 days squared
Step 5. Take the square root of the variance. The result will be the standard deviation
Answer on next slide
Measures of Spread – Standard Deviation
√43.5 = 6.59
Round to 6.6
The standard deviation is 6.6 days
Measures of Spread – Quartiles
Data can also be grouped into 4 equal parts, or QUARTILES
Each quartile includes 25% of the data
E.g., the cutoff for the first quartile is the 25th percentile
The cutoff for the second quartile is the 50th percentile
Third quartile cutoff = 75th percentile
Fourth quartile cutoff = 100th percentile
Quartiles
Method for determining the first (Q1) and third (Q3) quartiles
Step 1. Arrange the observations in increasing order
Step 2. Find the position of the 1st and 3rd quartiles with the following formulas. Divide the sum by the number of observations
Position of 1st quartile (Q1) = 25th percentile = (n + 1) / 4
Position of 3rd quartile (Q3) = 75th percentile = 3(n + 1) / 4
Step 3. Identify the value of the 1st and 3rd quartiles
If a quartile lies on an observation, the value of the quartile is the value of that observation
If a quartile lies between observations, the value of the is the value of the lower observation plus the specified fraction of the difference between observations – see the example on the next slide
Step 4. Report the values at Q1 and Q3
Quartiles
Example:
If there are 11 pieces of data (i.e., n = 11), which observation represent the 1st and 3rd quartiles?
Hint: use the formulas:
Q1 = (n + 1) / 4
And
Q3 = 3(n + 1) / 4
Answer on the next slide
Quartiles
The observation representing the 1st quartile is:
Q1 = (n + 1) / 4
Q1 = (11 + 1) / 4
12 / 4 = 3
Q3 = 3(n + 1) / 4
3(11 + 1)/ 4
3(12) / 4
36/ 4 = 9
The POSITION of the 1st quartile is 3 and the POSITION of the 3rd quartile is 9
Five-Number Summary
If our data are as follows (note that I’ve already placed them in numerical order), report the minimum, 1st quartile, median, 3rd quartile, and maximum
0, 2, 4, 4, 4, 5, 5, 6, 7, 8, 9
Answer of following slide
Five-Number Summary
0, 2, 4, 4, 4, 5, 5, 6, 7, 8, 9
Minimum = 0
1st quartile = 4
Median = 5
3rd quartile = 7
Maximum = 9
The End
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