1. Tukey Box Plots
Review

2. Range = highest value – lowest Value
Definitions:
Range: The range of a box plot is the spread of values of data points. The range should be expressed as:
Range = highest value – lowest Value
Example: Range = 15 – 1 = 14
NOTE: The Min and Max are not necessarily used to calculate range. The range is calculated by using the highest and lowest values.
Minimum (Min): The minimum is the lowest value within 1.5 IQR of the lower quartile (Q1).
Maximum (Max): The maximum is the highest value within 1.5 IQR of the higher quartile (Q3).
Median: The median is the data point or value in which 50% of the data is above the median and 50% of the data is below the median.
Q1 (1st Quartile): The Q1 is the data point or value in which 75% of the data is above the Q1 value and 25% of the data is below the Q1 value.
Inter-quartile Range (IQR): The IQR is the spread of values of data points between Q1 and Q3. The IQR should be expressed as:
IQR = Q3 value – Q1 Value
Example: IQR = 11 – 4 = 7
Q3 (3rd Quartile): The Q3 is the data point or value in which 25% of the data is above the Q3 value and 75% of the data is below the Q3 value.
Mean: The arithmetic average of all the data.

3. DESCRIPTIVE STATISTICS:
There are twelve (12) important descriptive statistics, some of which were on the previous slide. The twelve are listed below to the right in the order they are normally represented. Some definitions are listed below to the left. See the previous slide for the other definitions.
Observations (Frequency): This is simply the number of observations in the set.
Observations
Range
Median Q1 Q3
IQR
BIL
Outliers
Min(imum)
Max(imum)
Mean
Mode
Base Inter-quartile Limit (BIL): The BIL is calculated by multiplying the IQR by 1.5. The BIL is used to find possible outlier values.
Definition on previous slide.
Outliers: Outliers are data points that are distant from other observations, falling outside of the minimum and maximum for the set.
Mode: This is simply the data value that occurs most often. There can be more that one mode if two values that occur most often occur the same number of times. If no value occurs more than others, there is no mode.

4. DESCRIPTIVE STATISTICS EXAMPLE QITH TUKEY BOX PLOT:
Data set: { 5, 15, 3, 8, 7, 7, 9, 9, 8, 1, 13, 8, 7 }
Observations: 13
Range: 14
Median: 8
Q1: 6
Q3: 9
IQR: 3
BIL:
Outliers: 1, 15
Min(imum): 3
Max(imum): 13
Mean:
Mode: 7, 8
Original data set re-ordered
Data set: { 1, 3, 5, 7, 7, 7, 8, 8, 8, 9, 9, 13, 15 }

5. DESCRIPTIVE STATISTICS EXAMPLE WITH TUKEY BOX PLOT:
HOW DID WE DO THAT? (PART 1)
Observations: 13
Range: 14
Median: 8
Q1: 6
Q3: 9
IQR: 3
BIL:
Outliers: 1, 15
Min(imum): 3
Max(imum): 13
Mean:
Mode: 7, 8
Re-ordered data set: { 1, 3, 4, 5, 5, 6, 7, 8, 9, 9, 9, 13, 15 }
Range: Highest minus lowest: 15 – 1 = 14.
Median: Since there are 13 values, the center (middle)
value is the 7th one from either side, which is 8.
Q1: With 13 values, finding the Q1 means finding the
center value to the left of the median. In this case,
it falls between the 3rd and 4th values so, we have
to average those two values: ( ) = 12 , 12/2 = 6.
Q3: Again, with 13 values, finding the Q3 means finding
the center value, but this time to the right of the
median. In this case, it falls between the 9th and
10th values so, we have to average those two values:
( ) = 18 , 18/2 = 9.

6. DESCRIPTIVE STATISTICS EXAMPLE WITH TUKEY BOX PLOT:
HOW DID WE DO THAT? (PART 2)
Observations: 13
Range: 14
Median: 7
Q1:
Q3: 9
IQR:
BIL:
Outliers: 1, 15
Min(imum): 3
Max(imum): 13
Mean:
Mode: 9
Re-ordered data set: { 1, 3, 4, 5, 5, 6, 7, 8, 9, 9, 9, 13, 15 }
IQR: Q3 minus Q1: 9 – 6 = 3.
BIL: IQR times 1.5: = 4.5.
Outliers: This is the part with which a lot of people have
difficulty.
Upper Outlier: Q3 + BIL: = 13.5
Since 15 is higher than 13.5, 15 is an upper outlier.
Thirteen is less than 13.5, so it is not an outlier.
Lower Outlier: Q1 - BIL: = 1.5
Since 1 is lower than 1.5, 1 is a lower outlier.
Three is higher than 1.5, so it is not an outlier.

7. DESCRIPTIVE STATISTICS EXAMPLE WITH TUKEY BOX PLOT:
HOW DID WE DO THAT? (PART 2)
Observations: 13
Range: 14
Median: 7
Q1:
Q3: 9
IQR:
BIL:
Outliers: 1, 15
Min(imum): 3
Max(imum): 13
Mean:
Mode: 9
Re-ordered data set: { 1, 3, 4, 5, 5, 6, 7, 8, 9, 9, 9, 13, 15 }
Min: Since 1 is a lower outlier, the next value toward
the center from 1 is the minimum. The next value
toward the center is 3. The minimum of the set is 3.
Max: As 15 is a upper outlier, the next value toward
the center from 15 is the maximum. The next value
toward the center is 13. The maximum of the set is 13.
Mean: =94 94/13 =
Mode: The value, 9, appear three times in the set. No other
value appears this often. Thus, 9 is the mode.