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## Descriptive Statistics

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**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**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**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.**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**Variance**• 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**• 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