I. Mean and Median
The MEAN is the numerical average
of the data set.
Notes Unit 8: Mean,
Median, Standard
Deviation
The mean is found by adding all the
values in the set, then dividing the
sum by the number of values.
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Understanding Mean, Median and Standard Deviation: Calculation and Interpretation, Study notes of Statistics
An explanation of the concepts of mean, median, and standard deviation, along with step-by-step calculations and examples using test scores. Mean is the average value, median is the middle value, and standard deviation measures the spread of data from the mean.
What you will learn
- What is standard deviation and how is it calculated?
- How do you calculate the mean?
- How does standard deviation help in understanding the spread of data?
- What is the difference between mean and median?
- How do you find the median?
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I. Mean and Median
The MEAN is the numerical average
of the data set.
Notes Unit 8: Mean, Median, Standard Deviation The mean is found by adding all the values in the set, then dividing the sum by the number of values.
The MEDIAN is the number that is in the middle of a set of data
**1. Arrange the numbers in the set in order from least to greatest.
- Then find the number that is in the middle.**
Lets find Abby’s
MEAN science test
score?
÷
The mean is 87
63 73 84 86 88 95 97 97 100
The median is 88.
Half the numbers are less than the median. Half the numbers are greater than the median.
How do we find
the MEDIAN
when two numbers are in the middle?
**1. Add the two numbers.
- Then divide by 2.**
183 ÷ 2
The median is 91. Ex 2: Find the median.
B. Bell Curve: The bell curve, which represents a normal distribution of data, shows what standard deviation represents. One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.
C. Steps to Finding Standard Deviation
- Find the mean of the data.
- Subtract the mean from each value.
- Square each deviation of the mean.
- Find the sum of the squares.
- Divide the total by the number of items. 6)Take the square root.
Ex 1: Find the standard deviation
The math test scores of five students
are: 92,88,80,68 and 52.
- Find the mean : (92+88+80+68+52)/5 = 76.
- Find the deviation from the mean : 92 - 76= 88 - 76= 80 - 76= 68 - 76= - 8 52 - 76= - 24
- Square the deviation from the mean: 2
2
2
2
2 ( 24) 576
- Find the sum of the squares of the deviation from the mean: 256+144+16+64+576= 1056
- Divide by the number of data items: 1056/5 = 211.
Ex 2: Standard Deviation
A different math class took the
same test with these five test
scores: 92,92,92,52,52.
Find the standard deviation for
this class.
Remember:
- Find the mean of the data.
- Subtract the mean from each value.
- Square each deviation of the mean.
- Find the sum of the squares.
- Divide the total by the number of items. 6)Take the square root.
- Divide the sum of the squares by the number of items : 1920/5 = 384 variance
- Find the square root of the variance:
Thus the standard deviation of the second set of test scores is 19.6.
Consider both sets of scores. Both classes have the same mean, 76. However, each class does not have the same scores. Thus we use the standard deviation to show the variation in the scores. With a standard variation of 14.53 for the first class and 19.6 for the second class, what does this tell us?