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) Medianc) 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 x1, x2, …… xn observations with corresponding frequencies f1, f2, ……fn, then

x1 f1+ x2 f2+ ……. xn fn Σfx Mean = --------------------------------- = ---------- f1+ f2+ ……fn Σf Problem: Calculate the avg. no. of children / family from the

following data:

No. of Children (X) No. of families ( f ) Total no of children (fx)

0 30 0 x 30 = 01 52 1 x 52 = 522 60 2 x 60 = 1203 65 3 x 65 = 1954 18 4 x 18 = 725 10 5 x 10 = 506 5 6 x 5 = 30

Total = 240 = 519

Mean = 519/ 240 = 2.163

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, ------- = 4th 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) 4th + 5th 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.

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

Mode is 5

Bimodal - 2 and 6

No Mode

Merits DemeritsMean:

• 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

SymmetricData is symmetric if the left half of its

histogram is roughly a mirror of its right half.

SkewedData 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:

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

Mean deviation (MD) = _ Σ X - X = ------------ n

= 6 / 7 = 0.85

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 (σ2). [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)

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

S.D (σ ) = = Σ(X –X) 2 / n-1

=(√1924/ (12-1) _____= √174

= 13.2

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Range

4s

Estimation of Standard DeviationRange Rule of Thumb

x - 2s x x + 2s

Range 4sor

(minimumusual value)

(maximum usual value)

Range

4s =

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% within1 standard deviation

34% 34%

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

x - 2s x - s x x + 2sx + s

68% within1 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 + 3sx + s

68% within1 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% 0.1%

2.4% 2.4%

13.5% 13.5%

FIGURE 2-15