Graphical Methods and Data Description

• Characterizing or summarizing the nature of collections of data is important.
• A summary of a collection of data via a graphical display can provide insight regarding the system from which the data were taken.
• A Stem-and-leaf plot, a combined tabular and graphic display, can be used to study the behavior of the mass statistical data.
• Example: the following table show the life of 40 car batteries

  Figure 1.9: Table of Car Battery Life (in years).

  

Table 1.5: Stem (integer part)-and-Leaf (decimal part) Plot of Battery Life.

Stem	Leaf	Frequency
1	69	2
2	25669	5
3	0011112223334445567778899	25
4	11234577	8

Table 1.6: Double-Stem-and-Leaf Plot of Battery Life.

Stem	Leaf	Frequency
1.	69	2
2*	2	1
2.	5669	4
3*	001111222333444	15
3.	5567778899	10
4*	11234	5
4.	577	3

Table 1.7: Relative Frequency Distribution of Battery Life.

Class	Class	Frequency	Relative
Interval	Midpoint	$f$	Frequency
1.5-1.9	1.7	2	0.05
2.0-2.4	2.2	1	0.025
2.5-2.9	2.7	4	0.100
3.0-3.4	3.2	15	0.375
3.5-3.9	3.7	10	0.250
4.0-4.4	4.2	5	0.125
4.5-4.9	4.7	3	0.075

• By rotating a stem-and-leaf plot counter-clockwise through an angle of 90, the resulting columns of leaves form a picture that is similar to a histogram.

Figure 1.10: Relative frequency histogram.

• As the sample size becomes larger, the frequency histogram would approach a bell-shaped continuous probability distribution.

Figure 1.11: Estimating frequency distribution.

• Skewness of data. A distribution is symmetric if it can be folded along a vertical axis so that the two sides coincide, otherwise skewed.
• Relationship between the Mean, Median, and Mode;
  • For a symmetric histogram or frequency curve; mode = median = mean,
  • Skewed to the right; mode < median < mean,
  • Skewed to the left; mean < median < mode.

Figure 1.12: Skewness of data.

• Other distinguishing features of a sample. The distribution can be divided by computing percentiles of the distribution.
• These quantities give the analyst a sense of the tails of the distribution.
• Tails are the relatively extreme values, either small or large. For example,
  • the 95th percentile separates the highest 5% from the bottom 95%.
  • the 1st percentile separate the bottom 1% from the rest of the distribution.