Mean, median, and mode ug

Single value in series of observations which indicate the characteristics of observations All data / values clustered around it & used to compare between one series to another Measures: a) Mean ( Arithmetic / Geometric / Harmonic) b) Median c) Mode

It is sum of all observations divided by number of observations __ Σx Mean ( X ) = ------ ( x= observation & n= no of observations) n Problem : ESR of seven subjects is 8,7, 9, 10, 7, 7 and 6. Calculate the mean. 8+7+9+10+7+7+6 54 Mean= -------------------------- = ------- = 7.7 7 7

For discrete observation: If we have x 1 , x 2 , …… x n observations with corresponding frequencies f 1 , f 2 , ……f n, then x 1 f 1 + x 2 f 2 + ……. x n f n Σfx Mean = --------------------------------- = ---------- f 1 + f 2 + ……f n Σf Problem : Calculate the avg. no. of children / family from the following data: Mean = 519/ 240 = 2.163 No. of Children (X) No. of families ( f ) Total no of children (fx) 0 30 0 x 30 = 0 1 52 1 x 52 = 52 2 60 2 x 60 = 120 3 65 3 x 65 = 195 4 18 4 x 18 = 72 5 10 5 x 10 = 50 6 5 6 x 5 = 30 Total = 240 = 519

When observations are arranged in ascending or descending order of magnitude, the middle most value is known as Median Problem : Same example of ESR as in mean observations are arranged first in ascending order, i.e 6, 7, 7, 7, 8, 9, 10 n+1 7+1 When n is odd, Median = ------ th observation i.e, ------- = 4 th observation = 7 2 2 n/2 th + (n/2 +1) th observation When n is even , Median = ---------------------------------------------- 2 So, if there are 8 observations of ESR like 5, 6, 7, 7,7, 8, 9, 10 n/2 th + (n/2 +1) 4 th + 5 th 7+ 7 Median = ------------------------th observation = ----------------th observation = --------- = 7 = 2 2 2

The mode is the data item that appears the most. If all data items appear the same number of times, then there is no mode.

5, 4, 6, 11, 5, 7, 10, 5 The mode is 5.

Mode is 5 Bimodal - 2 and 6 No Mode a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Examples

Merits Demerits Mean: Rigidly defined Based on all observations Easy to calculate & understand Least affected by sampling fluctuation, hence more stable Mean: Can be used only for quantitative data Unduly affected by extreme observations Median: Not affected by extreme observations Both for quantitative & qualitative data Median: Affected more by sampling fluctuations Not rigidly defined Can be used for further mathematical calculation Mode: Not affected by extreme observations Both for quantitative & qualitative data Mode: Not rigidly defined Can be used for further mathematical calculation

Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half. Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Definitions

Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)

Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)

Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-13 (b) Figure 2-13 (a) Figure 2-13 (c)

Biological variation in large groups is common. e.g : BP, wt What is normal variation? and How to measure? Measure of dispersion helps to find how individual observations are dispersed around the central tendency of a large series Deviation = Observation - Mean

Range Quartile deviation Mean deviation Standard deviation Variance Coefficient of variance : indicates relative variability ( SD/Mean) x100

Range : difference between the highest and the lowest value Problem: Systolic and diastolic pressure of 10 medical students are as follows: 140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 & 154/90. Find out the range of systolic and diastolic blood pressure Solution: Range of systolic blood pressure of medical students: 90-160 or 70 Range of diastolic blood pressure of medical students: 60-90 or 30 Mean Deviation: average deviations of observations from mean value _ Σ (X – X ) __ Mean deviation (M.D) = --------------- , ( where X = observation, X = Mean n n= number of observation )

Problem : Find out the mean deviation of incubation period of measles of 7 children, which are as follows: 10, 9, 11, 7, 8, 9, 9. Solution: Mean deviation (MD ) = _ Σ X - X = ------------ n = 6 / 7 = 0.85 Observation (X) __ Mean ( X ) __ Deviation (X - X) 10 __ X = Σ X / n = 63 / 7 = 9 1 9 0 11 2 7 -2 8 -1 9 0 9 0 ΣX=63 _ Σ (X-X) = 6, ignoring + or - signs

It is the most frequently used measure of dispersion S.D is the Root-Means-Square-Deviation S.D is denoted by σ or S.D ___________ Σ ( X – X ) 2 S.D (σ) = γ ---------------------- n

Calculate the mean ↓ Calculate difference between each observation and mean ↓ Square the differences ↓ Sum the squared values ↓ Divide the sum of squares by the no. observations (n) to get ‘mean square deviation’ or variances (σ 2 ). [For sample size < 30, it will be divided by (n-1)] ↓ Find the square root of variance to get Root-Means-Square-Deviation or S.D ( σ)

S.D ( σ ) = = Σ(X –X) 2 / n-1 =(√1924/ (12-1) _____ = √174 = 13.2 Observation (X) __ Mean ( X ) _ Deviation (X- X) __ (X-X) 2 58 __ X = Σ X / n = 984/12 = 82 -12 576 66 -16 256 70 -12 144 74 -8 64 80 -2 4 86 -4 16 90 8 64 100 18 324 79 -3 9 96 14 196 88 6 36 97 15 225 Σ X = 984 _ Σ (X - X) 2 =1914

Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value)

Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s 

Estimation of Standard Deviation Range Rule of Thumb x - 2 s x x + 2 s Range  4 s or (minimum usual value) (maximum usual value) Range 4 s  = highest value - lowest value 4

minimum ‘usual’ value  (mean) - 2 (standard deviation) minimum  x - 2(s)

minimum ‘usual’ value  (mean) - 2 (standard deviation) minimum  x - 2(s) maximum ‘usual’ value  (mean) + 2 (standard deviation) maximum  x + 2(s)

x The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

x - s x x + s 68% within 1 standard deviation 34% 34% The Empirical Rule (applies to bell-shaped distributions ) FIGURE 2-15

x - 2s x - s x x + 2s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations The Empirical Rule (applies to bell-shaped distributions ) 13.5% 13.5% FIGURE 2-15

x - 3s x - 2s x - s x x + 2s x + 3s x + s 68% within 1 standard deviation 34% 34% 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean The Empirical Rule (applies to bell-shaped distributions ) 0.1% 2.4% 2.4% 13.5% 13.5% FIGURE 2-15 0.1%

Mean, median, and mode ug