1. Biostatistics
Unit 3 - Graphs

2. Grouped data
Data can be grouped into a set of non-overlapping, contiguous intervals called class intervals (bins).  Class intervals are used to sort the data.  Between 6 and 15 class intervals are usually used depending on the range of the data.

3. Grouped data
The frequency tells how many of the data values fall into each class interval.  Frequency can be displayed graphically using the histogram and the frequency polygon.

4. Bacterial cell lengths
Below are the measured lengths of 30 individual bacterial cells.  As they have not yet been sorted to make a sorted list, they can be considered as raw data.
1)   1.5          2)   2.0          3)   2.0          4)   3.0          5)   2.0  
6)   3.2          7)   2.3          8)   1.5          9)   2.0        10)   2.0
11)   1.0        12)   1.0        13)   2.5        14)   3.4        15)   2.1
16)   2.0        17)   4.0        18)   3.0        19)   2.0        20)   2.0
21)   2.2        22)   2.0        23)   2.0        24)   2.0        25)   2.0
26)   1.5        27)   2.0        28)   1.0        29)   1.0        30)   1.0

5. Basic statistics
The values of the basic statistics for the data are presented below.  They were obtained using the TI-83 calculator.  Similar results are available using Microsoft Excel.     n = 30                min x = 1.0     mean = 2.04       Q1 = 1.5     s = .7186            median = 2.0     mode = 2.0         Q3 = 2.2     range = 3.0         max x = 4.0

6. Frequency Table for Bacterial Cell Lengths
Class interval ( m )    Frequency
                      5                          19                            5                            1

7. Histogram

8. Frequency Polygon

9. Percentiles and quartiles
Percentiles are used for location of data on the horizontal axis.  The median corresponds to the 50th percentile.  We generally are interested in quartiles which are 25th percentiles.  The first quartile (Q1) is the 25th percentile.  It contains one-quarter of the data.  The second quartile is the median which marks the point with half of the data.
(continued)

10. Percentiles and quartiles
The third quartile (Q3) is the 75th percentile representing three-quarters of the data.  Using our ordered observations, the quartiles are calculated using the formulas below.  
The interquartile range is represented by Q3 - Q1.

11. Calculation of quartiles
For the data set of 30 observations of bacteria, this means that: Q1 = (n+1)/4 -> > 8th observation (1.5) Q2 = 2(n+1)/4 -> > average of the 15th and 16th observations (2.0) Q3 = 3(n+1)/4 -> > 23rd observation (2.2)
(continued)

12. Calculation of quartiles
Be careful when interpreting quartile calculations.  An answer of Q1 = 7.75 rounded to 8 does not mean that the first quartile is 8.  It means that the first quartile is the data item in the 8th position.  In this data set that value is 1.5.

13. Box Plot
The box plot is used to convey information about the data.  It makes use of the quartiles that were calculated above.
1. Draw a number line representing cell length on the horizontal axis. 2.    Above the horizontal axis draw a rectangle with the left-hand end of the rectangle directly above Q1 and the right-hand end of the rectangle directly above Q3.

14. Box Plot
3.    Draw a vertical line across the box directly over Q2. 4.    Draw a horizontal extension line out of the left-hand end of the box to a point above the smallest measurement of the data.  At this point draw a vertical line. 5.    Draw another horizontal line out of the right-hand end of the box to a point above the largest measurement of the data.  Draw a vertical line at this point.

15. Box plot of cell measurements

16. fin