SUMMARIZING DATA: NUMERICAL MEASURES, Lecture notes of Mathematical Statistics

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SUMMARIZING DATA: NUMERICAL MEASURES

WEEK 3

• MEASURES OF CENTRAL TENDENCY-

SUMMARIZING DATA Organizing data ü a) Tables ü b) Diagrams c) Numerical Measures

THE Σ ( SIGMA) SIGN Dr. Doğukan ÖZEN 32 The sign Σ ( sigma) is a summation sign. We can write (x 1 +x 2 +x 3 +… xn) as --> If a and b are integers and a < b, then; means à xa+xa+1+xa+2+… xb Question : If x 1 = 3, x 2 = 6 and x 3 =^ - 5 then find the following? a) (^) b) c)

• Most widely used measure of central tendency !!
• Arithmetic mean is the sum of all observations divided by the number of observations.
• In statistical terms, it can be written as à ARITHMETIC MEAN

MEDIAN

• An alternative measure to mean. (More precisely: sample median!)
• Suppose there are n observations in a sample. If these observations are

ordered from smallest to largest, then the median can be defined as;

% &

th largest observation if n is odd

The average of th^ and^ th^ largest observations if n is even.

EXAMPLE:

The following table consist of somatic cell count measurement (x10000) of milk samples taken

from 10 Holstein in a dairy farm. Compute the median value of somatic cell count.

i xi i xi 1 11 6 8 2 21 7 9 3 18 8 110 4 14 9 12 5 13 10 20

Solution:

Step 1. Order the sample from smallest to largest.

Step 2. Because n is even (n=10), sample median is the

average of 5

th

and 6

th

observations.

Step 3. Compute the sample median = (13+14)/2= 13,

THE GEOMETRIC MEAN

• Some of the laboratory data can be expressed either as multiples of 2 or as a constant multiplied by a power of 2.
• So the outcomes can be in a form of 2 k c, where k=0,1,2,3,… (with a constant c) A possible solution can be by using log-transformed observations and then taking the arithmetic mean of the observations: Geometric mean =

EXAMPLE

• Compute the geometric mean of 3, 5, 6, 6, 7, 12 and 20. Arithmetic mean = 8. Median = 6 Mode = 6

Median

• widely used for markedly skewed distributions because it is sensitive only to the number rather than to the values of scores above and below it
• the most stable measure that can be used with open-ended distributions
• more subject to sampling fluctuation than the mean

Mode

• more appropriate than the mean or the median for quantitative variables that are inherently discrete
• the only measure appropriate for unordered qualitative variables
• much more subject to sampling fluctuation than the mean and the median A SUMMARY OF MEAN, MEDIAN AND MODE

LOCATION OF MEAN, MEDIAN AND MODE IN A DISTRIBUTION To skew means to stretch in one direction. A distribution is skewed to the left if the left tail is longer than the right tail. A distribution is skewed to the right if the right tail is longer than the left tail. A left-skewed distribution stretches to the left, a right-skewed to the right. 42 The Normal Curve: represents the symmetrical distribution Negative direction Positive direction Negatively skewed No skew Positively skewed Mode Median Mean Mode Median Mean Mean Median Mode Frequency

• MEASURES OF DISPERSION-

SUMMARIZING DATA: NUMERICAL MEASURES

• Let’s say two different group of investigators collected the following data:
  • Group 1: 30, 120, 130, 80, 90
  • Group 2: 88, 92, 90, 86, 94

Mean = 90

To describe the data, we also need to give information about the dispersion!

MEASURES OF DISPERSION

• Defined as the distance between the largest and the smallest observations = (Max-Min) RANGE (^) ( R )

Range;

can overestimate the dispersion due to extreme values.

Tends to increase in value as the number of observations in the sample increases.

INTERQUARTILE RANGE

• It is the difference between the first (25% - > Q1) and third quartiles (75% - > Q3).

The interquartile range;

Ø Is not influenced by extreme values or sample size

Ø Can be misleading due to ignoring most of the observations (it is calculated from

only two of points)