Most frequency distributions exhibit a central tendency ie:- a shape such that the bulk of the observations pile up in the area between 2 extremes. There are 3 principal measures of central tendency:-

1.Mean2.Median3.Mode

MeanThis is calculated by adding the observations

and dividing by the number of observations.

For example, number of patients treated on 8 days.

ArithmeticMean

Day No. 1 2 3 4 5 6 7 8

No of patients

86 52 49 42 35 31 30 11

428

1131313542495286

MedianIt is the middle most or most critical value

when figures are arranged according to size. Half of the items lie above this point and half lie below it. It is used for reducing the effects of extreme values or few data which can be ranked but not economically measurable (Eg. Shades of colour)

If the data in an even number of items, median is the average of the 2 middle items.

Median = The th item in a data array where n is the no. of items in the array.

2

1n

ExamplesData set with odd no. of items

Data set with even no. of items

Item

1 2 3 4 5 6 7

Time

4 4.3

4.7

4.8

5 5.1

6.0

# 1 2 3 4 5 6 7 8

No

86 52 49 43 35 31 30 11

Median

Median = th item = th item 2

1n5.4

2

18

Median = 392

3543

ModeIt is the value which occurs most often in data

set.It is the value which is used for severely

skewed distributions, describing irregular situations where 2 peaks are found or for eliminating the effect of extreme values.

Eg. No. of delivery trips made per day made by an RMC plant.0 2 5 7 15

0 2 5 7 15

1 4 6 8 15

1 4 6 12 19

Modal value is 15 because it

occurs most often.

The modal value 15 inplies that the plant activity is higher than 6.7 (which is mean). The mode tells us that 15 is the most frequent no. of trip, but it fails to let us know that most of the values are under 10.

A distribution in which the values of mean, median and mode coincide is known as symmetrical distribution. When they do not coincide, the distribution is known as skewed or asymmetrical.

If distribution is moderately skewed, Mean – Mode = 3(Mean-Median)

DispersionThe measure of dispersion or scatter are

Range R and sample standard deviation and variance

s

ExamplesSample 1 Sample 2

Larger scatter

Normal curve and its significance

Much of the variation in nature and industry follows the normal curve (Gaussian curve). It is the bell shaped, symmetrical form as above. Although most of the area is covered within the limits , the curve extends from -∞ to +∞.

Variation in height of human beings, variation of weight of elements, life of 60W bulbs etc are expected to follow the normal curve.

Limits % of total area within specified limits

The test scores of a sample of 100 students have a symmetrical distribution with a mean score of 570 & standard deviation of 70, approx. What scores are between a) 430 and 710b) 360 and 780

a) Hence 95% approx. have scores between 430 &

710.

b)Hence 99.7% of scores are between 360 & 780.

Normal distribution table gives to 4 decimal places the proportion of the total area under the normal curve that occurs between -∞ and +∞ expressed in multiplying σ on either side of μ. It can be use to find out the area between any 2 chosen points.

Eg. Area between and Table A reading for +1 σ = 0.8413Table A reading for -1.75 σ = 0.0401Area enclosed = 0.2012The mathematical equation for the normal curve is given by

Where

Mathematically, Table is described by

Thus the values read from table represents the area under the curve from - ∞ to z.

Central Limit TheoremFrom the standpoint of process control, the central

limit theorem is a powerful tool.“Irrespective of the sample of the distribution of a

universe, the distribution of average values, of a subgroup of size , drawn from the universe will tend to a normal distribution as the subgroup size ‘n’ grows without bound.”

If simple random sample sizes ‘n’ are taken from a population having a mean μ and standard deviation σ, the probability distribution with mean μ and

Standard deviation as ‘n’ becomes large. ”

The power of the Central Limit Theorem can be seen through computer simulation using the quality game box software.

If is the mean of sample size ‘n’ taken from a population having the mean μ and variance , then

is a random variable whose distribution approaches that of the standard normal distribution as

Central Limit Theorem simulation on Quality Gamebox

ExampleIf one litre of paint covers on an average, 106.7

sq.m of surface with a standard deviation of 5.7 sq.m, what is the probability that the sample mean area covered by a sample of 40g that 1 litre cans will be anywhere from 100 to 110 sq.m (using Central Limit Theorem)

By the central limit theorem, we can find the area between in normal curve.

For

then area between 100 to 110 sq.m = 1-0 = 1ie:- 100% will be covered

Deming’s funnel experimentIn this experiment, a funnel in suspended above a

table with a target drawn on a table with a tablecloth. The goal is to hit the target. Participants drop a marble through the funnel and mark the place where the marble eventually lands. Rarely does the marble rest on the target. The variation is due to common causes in the process. One strategy is to simply leave the funnel alone, which creates some variation of points around the target. This may be called Rule 1. However, many people believe they can improve the result by adjusting the location of funnel. 3 possible rules for adjusting funnel are:-

Rule 2: Measure deviation from the point at which marble comes to rest and the target. Move the funnel an equal distance in opportunities from its current position. (Fig for Rule 2)

Rule 3: Measure deviation from the point at which the marble comes to root and the target. Set the funnel an equal distance in the opposite direction of error from target. (Fig of Rule 3)

Rule 4: Place the funnel over the spot where the marble last came to rest. Fig. shows a computer simulation of these strategies using Quality Gamebox.

People use these rules inappropriately all the time, causing more variation than would normally occur. An amateur golfer who hits bad shots tends to make an immediate adjustment.

The purpose of this experiment is to show that people can an do affect the outcome of many processes and create unwanted variation by ‘tampering’ with the process or indiscriminately trying to remove common/ chance causes of variation.