Basic Data Analysis

1
Basic Data Analysis
2
Measures of Central Tendency

• Mode
• Median
• Mean

3
The Mode

• the value or property that occurs most frequently
  in the data

4
Find the mode

• 6, 7, 2, 3, 4, 6, 2, 6
• The mode is 6.

5
Find the mode

• 6, 7, 2, 3, 4, 5, 9, 8
• There is no mode for this data.

6
The Median

• the central value of an ordered distribution

7
To find the median of raw data

• Order the data from smallest to largest.
• For an odd number of data values, the median is
  the middle value.
• For an even number of data values, the median is
  found by dividing the sum of the two middle
  values by two.

8
Find the median

• Data 5, 2, 7, 1, 4, 3, 2
• Rearrange 1, 2, 2, 3, 4, 5, 7

The median is 3.
9
Find the median
Data 31, 57, 12, 22, 43, 50 Rearrange 12, 22,
31, 43, 50, 57
The median is the average of the middle two
values
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The Mean

• The mean of a collection of data is found by
• summing all the entries
• dividing by the number of entries

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Find the mean
6, 7, 2, 3, 4, 5, 2, 8
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Sigma Notation

• The symbol ? means sum the following.
• ? is the Greek letter (capital) sigma.

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Notations for mean

• Sample mean
• x bar

• Population mean
• Greek letter (mu)

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Number of entries in a set of data

• If the data represents a sample, the number of
  entries n.
• If the data represents an entire population, the
  number of entries N.

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Sample mean
16
Population mean
17
Quartiles

• Percentiles that divide the data into fourths
• Q1 25th percentile
• Q2 the median
• Q3 75th percentile

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Quartiles
Median Q2
Q1
Q3
Lowest value
Highest value
Inter-quartile range IQR Q3 Q1
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Computing Quartiles

• Order the data from smallest to largest.
• Find the median, the second quartile.
• Find the median of the data falling below Q2.
  This is the first quartile.
• Find the median of the data falling above Q2.
  This is the third quartile.

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Find the quartiles

• 12 15 16 16 17 18 22 22
• 23 24 25 30 32 33 33 34
• 41 45 51

The data has been ordered. The median is 24.
21
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
The data has been ordered. The median is 24.
22
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
For the data below the median, the median is
17. 17 is the first quartile.
23
Find the quartiles
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
For the data above the median, the median is
33. 33 is the third quartile.
24
Find the interquartile range

• 12 15 16 16 17 18 22 22
• 23 24 25 30 32 33 33 34
• 41 45 51

IQR Q3 Q1 33 17 16
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Five-Number Summary of Data

• Lowest value
• First quartile
• Median
• Third quartile
• Highest value

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Box-and-Whisker Plots

• a graphical presentation of the five-number
  summary of data

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Making a Box-and-Whisker Plot

• Draw a vertical scale including the lowest and
  highest values.
• To the right of the scale, draw a box from Q1 to
  Q3.
• Draw a solid line through the box at the median.
• Draw lines (whiskers) from Q1 to the lowest and
  from Q3 to the highest values.

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Construct a Box-and-Whisker Plot
12 15 16 16 17 18 22 22 23 24 25 30
32 33 33 34 41 45 51
Lowest 12 Q1 17 median 24 Q3 33 Highest
51
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Box-and-Whisker Plot
Lowest 12 Q1 17 median 24 Q3 33 Highest
51
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Resistant Measure

• a measure that is not influenced by extremely
  high or low data values

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Which is less resistant?

• Mean
• Median

• The mean is less resistant. It can be made
  arbitrarily large by increasing the size of one
  value.

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Measures of Variation

• Range
• Standard Deviation
• Variance
• Mean Absolute Deviation (MAD) in GPS

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The Range

• the difference between the largest and smallest
  values of a distribution

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Find the range

• 10, 13, 17, 17, 18
• The range largest minus smallest
• 18 minus 10 8

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The standard deviation

• a measure of the average variation of the data
  entries from the mean

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Standard deviation of a sample
mean of the sample
n sample size
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To calculate standard deviation of a sample

• Calculate the mean of the sample.
• Find the difference between each entry (x) and
  the mean. These differences will add up to zero.
• Square the deviations from the mean.
• Sum the squares of the deviations from the
  mean.
• Divide the sum by (n ? 1) to get the variance.
• Take the square root of the variance to get the
  standard deviation.

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The Variance

• the square of the standard deviation

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Variance of a Sample
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Find the standard deviation and variance
x 30 26 22
4 0 ?4
16 0 16 ___
Sum 0
78
32
mean 26
41
The variance
32 ? 2 16
42
The standard deviation
s
43
Find the mean, the standard deviation and variance
x 4 5 5 7 4
?1 0 0 2 ?1
1 0 0 4 1
mean 5
25
6
44
The mean, the standard deviation and variance
Mean 5
45
Population Mean and Standard Deviation
46
MEAN ABSOLUTE DEVIATIONMAD (Added to GPS)
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To Find Mean, Median, Quartiles, and Standard
Deviation On TI-83/84
Calculator 1) Place the data in L1 (Press STAT
EDIT to see L1, L2 etc) 2) press the STAT
button 3) Move cursor to Calc 4) Press enter 5)
Press L1 and press enter For Mean read
Standard Deviation of Sample read
Standard Deviation of Population read
Scroll down to read Min, Q1, Med, Q3, and
Max