Descriptive Statistics

Anwar Ahmad

Central Tendency- Measure of location

• Measures descriptive of a typical or representative value in a group of observations

• It applies to groups rather than individuals

Arithmetic Mean

• Simplest and obvious measure of central tendency

• Simple average of the observations in the group, i.e. the value obtained by adding the observations together and dividing this sum by the number of observations in the group

Arithmetic Mean

Example:

4,5,9,1,2

21/5

4.2

n

i

in xn

xxxn

x1

1121

Median

• The middle value in a set of observations ordered by size

• Median income or median house price

• 1,2,4,5,9

• 4 is the median

Mode

• The most frequently occurring value in a set of observations.1,2,2,4,5,9

• 2 is the mode

Other Measures of Central Tendency

• Midrange: The value midway between the smallest and largest values in the sample, that is, the arithmetic mean the largest and smallest values, the extremes.

• 4,5,9,1,2

• (9+1)/2

• 5

Geometric Mean

• The geometric mean of a set of observations is the nth root of their product.

• Gm of 4 & 9• Sqrt 4*9• Sqrt 36• 6

ixi1n

Harmonic Mean

• The harmonic mean of a set of observations is the reciprocal

(1/x) of the arithmetic mean of the reciprocals of the

observations.

ni

1

xi

Harmonic Mean

• Av. Velocity of car that traveled first 10 mi. at 30 mph; and the second 10 mi. at 60 mph.

• Mean 30+60 /2 = 45 ?

• Total distance by total time

• 10+10 / 1/3 + 1/6 hr (1/2 hr)

• 20/ ½ hr

• Av. velocity 40 mph

Harmonic Mean

• Harmonic mean

• 2/ (1/30+1/60) = 40

Weighted Mean

• When all observations do not have equal weight

• Lab A 50 cultures, 25 positive, 50%• Lab B 80 cultures, 60 positive, 75%• Lab C 120 cultures, 30 positive, 25% =

150/3 =50%• WM = 50(50%)+80(75%)+120(25%) /

50+80+12• 46%

Measure of Variability

• 1,4,4,4,7 = 20 = 20/5 = 4 variation

• 4,4,4,4,4 = 20 = 20/5 = 4 no variation

• Same means, median, mode

• 0 if no variation

• Some + value, if there is a variation

• Variation from the mean

Measure of Variability

• Range

• Variance

• Standard Deviation

Range • Range is the simplest measure of spread or

dispersion: • It is the difference between the largest and

the smallest values. • The range can be a useful measure of spread

because it is easily understood. • However, it is very sensitive to extreme

scores since it is based on only two values.

Range

• The range should almost never be used as the only measure of spread, but can be informative if used as a supplement to other measures of spread such as the standard deviation or variance

Variance

• Squared deviation from the mean.

• 1,4,4,4,7, mean 4• (1-4), (4-4), (4-4), (4-4), (7-4)• -3, 0, 0, 0, 3 = 0• -32, 0, 0, 0, 32 = 18/5 = 18/4 =

4.5

1

n1

ixi x2

• The variance describes the heterogeneity of a distribution and is calculated from a formula that involves every score in the distribution. It is typically symbolized by the letter s with a superscript "2". The formula is

Variance, s2 = sum (scores - mean)2/(n - 1) degree of freedom

Variance

Variance

• The variance is a measure of how spread out a distribution is. It is computed as the average squared deviation of each number from its mean.

Standard deviation

• The square root (the positive one) of the variance is known as the "standard deviation." It is symbolized by s with no superscript.

• Sqrt 4.5• 2.12

1

n1

ixi x2

Summary Formulae