1. Unit-3
Population—
In statistics, population is the total set of observations thatcan be made.
For example,
1. If weare studying the weight of adult women, the population is the set of
weights of all the women in the world.
2. If we are studying the grade point average(GPA) of students at Harvard,
the population is the set of GPA's of all the students at Harvard.
Sampling—
sampling is concerned with the selection of a subsetof individuals fromwithin a
statistical population to estimate characteristics of the whole population.
Populationv/s Sample
The main differencebetween a population and samplehas to do with how
observations areassigned to the data set.
A population includes each element fromthe set of observations that can
be made.
A sample consists only of observations drawn fromthepopulation.
Depending on the sampling method, a sample can have fewer observations
than the population, the samenumber of observations, or more
observations.
More than one sample can be derived fromthe samepopulation.
Other differences haveto do with nomenclature, notation, and computations. For
example,
A measurablecharacteristic of a population, such as a mean or standard
deviation, is called a parameter; buta measurable characteristic of a
sample is called a statistic.
2. The mean of a population is denoted by the symbolμ; but the mean of a
sample is denoted by the symbol 𝑋.
The formula for the standard deviation of a population is different from
the formula for the standard deviation of a sample.
Random Sampling:-
Random sampling is a sampling technique where we select a group of subjects (a
sample) for study from a larger group (a population). Each individual is chosen entirely
by chance and each member of the population has a known, but possibly non-equal,
chance of being included in the sample.
Simple Random Sampling
Simple random sampling is the basic sampling technique where we select a group of
subjects (a sample) for study from a larger group (a population). Each individual is
chosen entirely by chance and each member of the population has an equal chance of
being included in the sample. Every possible sample of a given size has the same
chance of selection; i.e. each member of the population is equally likely to be chosen at
any stage in the sampling process.
More precisely,
A sampling method is a procedurefor selecting sample elements froma
population. Simple random sampling refers to a sampling method that has the
following properties.
The population consists of N objects.
The sample consists of n objects.
All possiblesamples of n objects are equally likely to occur.
An important benefit of simple randomsampling is that it allows researchers to
use statistical methods to analyzesample results.
For example, given a simple randomsample, researchers can usestatistical
methods to define a confidence interval around a sample mean.
Statistical analysis is not appropriate when non-randomsampling methods are
used.
3. There are many ways to obtain a simple randomsample.
One way would be the lottery method. Each of the N population members is
assigned a unique number. The numbers areplaced in a bowland thoroughly
mixed. Then, a blind-folded researcher selects n numbers. Population members
having the selected numbers areincluded in the sample.
Standard Error
The standard error (SE) is the standard deviation of the sampling distribution of a
statistic.The term may also be used to refer to an estimate of that standard
deviation, derived froma particular sample used to compute the estimate.
For example, the sample mean is the usualestimator of a population mean.
However, differentsamples drawn fromthat samepopulation would in general
have different values of the sample mean, so there is a distribution of sampled
means (with its own mean and variance).
The standarderror of the mean (SEM) (i.e., of using the sample mean as a
method of estimating the population mean) is the standard deviation of those
sample means over all possiblesamples (of a given size) drawn fromthe
population. Secondly, the standard error of the mean can refer to an estimate of
that standard deviation, computed fromthe sampleof data being analyzed at the
time.
1. The smaller the standard error, the morerepresentative the sample will be
of the overall population.
2. The standard error is also inversely proportionalto the sample size; the
larger the sample size, the smaller the standard error becausethe statistic
will approach the actual value.
The standard error of the mean, also called the standard deviation of the mean, is
a method used to estimate the standard deviation of a sampling distribution.
To understand this, first weneed to understand why a sampling distribution is
required.
Example:-
4. Consider an experiment that measures the speed of sound in a material along the
three directions (along x, y and z coordinates). By taking the mean of these values,
we can get the averagespeed of sound in this medium.
However, there are so many external factors that can influence the speed of
sound, like small temperature variations, reaction time of the stopwatch, pressure
changes in the laboratory, wind velocity changes, and other randomerrors. Thus
instead of taking the mean by one measurement, we prefer to take several
measurements and take a mean each time. This is a sampling distribution. The
standard error of the mean now refers to the change in mean with different
experiments conducted each time.
Mathematically, the standard error of the mean formula is given by:
𝜎 𝑀 =
𝜎
√ 𝑁
𝜎𝑀 = standard error of the mean
𝜎 = the standard deviation of the original distribution
𝑁 = the sample size
√ 𝑁 = Root of the sample size
Itcan be seen from the formula that the standard error of the mean decreases as
𝑁 increases. This is expected because if the mean at each step is calculated using
a lot of data points, then a small deviation in one value will causeless effect on
the final mean.
The standard error of the mean tells us how the mean varies with different
experiments measuring the samequantity. Thus if the effect of randomchanges
are significant, then the standard error of the mean will be higher. If there is no
change in the data points as experiments are repeated, then the standard error of
mean is zero.
Standard Error of the Estimate
5. A related and similar concept to standard error of the mean is the standard error
of the estimate. This refers to the deviation of any estimate from the intended
values.
For a sample, the formula for the standard error of the estimate is given by:
𝑆𝑒𝑠𝑡 = √
∑( 𝑌 − 𝑌′)2
𝑁 − 2
where 𝑌 refers to individual data sets, 𝑌′ is the mean of the data and 𝑁 is the
sample size.
Note that this is similar to the standard deviation formula, but has an 𝑁 − 2 in the
denominator instead of 𝑁 − 1 in caseof sample standard deviation.
STANDARD ERROR CALCULATION
Procedure:
Step 1: Calculate the mean (Total of all samples divided by the
number of samples).
Step 2: Calculate each measurement's deviation from the mean
(Mean minus the individual measurement).
Step 3: Square each deviation from mean. Squared negatives become
positive.
Step 4: Sum the squared deviations (Add up the numbers from step3).
Step 5: Divide that sum from step 4 by one less than the sample size
(n-1, that is, the number of measurements minus one)
Step 6: Take the square root of the number in step 5. That gives you
the "standard deviation (S.D.)."
6. Step 7: Divide the standard deviation by the square root of the sample
size (n). That gives you the
“standard error”.
Step 8: Subtract the standard error from the mean and record that
number. Then add the standard error to the mean and record that
number. You have plotted mean ±1 standard error (S. E.),
the distance from 1 standard error below the mean to 1 standard error
above the mean.
Example:
Name Height to nearest
0.5 cm
2 Deviations (𝜇 −
𝑖)
3 Squared
deviations (𝜇 − 𝑖)2
1. Waldo 150.5 11.9 141.61
2. Finn 170.0 -7.6 57.76
3. Henry 160.0 2.4 5.76
4. Alfie 161.0 1.4 1.96
5. Shane 170.5 -8.1 65.61
N=5 1
Mean 𝜇= 162.4 cm
4Sum of squared
deviations
∑(𝜇 − 𝑖)
2
= 272.70
5 Divide by number of measurements-1.
∑(𝜇 − 𝑖)
2
𝑛 − 1
=
272.70
4
= 68.175
6 Standard deviation = √
∑(𝜇−𝑖)2
𝑛−1
= √68.175 = 8.257
7. 7 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟 =
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
√ 𝑛
=
8.257
2.236
= 3.69
8 𝜇 ± 1𝑆𝐸 = 162 ± 3.7 or 159cm to 166cm for the men (162.4 − 3.7 𝑡𝑜 162.4 +
3.7).
Example
This shows four samples of increasing size. Note how the standard error reduces with increasing
sample size.
Sample 1 Sample 2 Sample 3 Sample 4
9 6 5 8
2 6 3 1
1 8 6 7
8 4 1
3 7 3
8 2 3
6 4
9 7
7 1
1 8
1 9
7 9
3
1
6
8
3
