Introduction of Random sampling
A sample is a part of the population. You can gain information about a population by examining samples of the population.
There are two types of samples
- Unbiased sample
- Biased sample
- An unbiased sample is representative of the population. It is chosen at random and is large enough to provide reliable information.
- A biased sample is not representative of a population. One or more groups of people are given benefits.
- An unbiased sample’s results are proportional to the population’s results. As a result, unbiased samples can be used to draw conclusions about a population. Samples that are skewed are not representative of the population. So, you should not use them to make conclusions about the population.
- You can use unbiased samples to make conclusions about populations. Different samples often have slightly different conclusions due to variability in the sample data.
What is Sampling?
Sampling is a central concept in statistics. Examining every element in a population is usually impossible. As a result, research and media articles frequently refer to a “sample” of a population. A graph of sample proportions for many different samples is the sampling distribution of the sample proportion.
What is Random Sampling?
Definition: A random sample is one where every element in the set has an equal chance of being selected. When people select a sample they believe will be random, it is usually not representative of a true random sample. Random samples are usually similar to the population.
1. Random samples from the same population will vary from sample to sample. Each element has an equal chance of being chosen, and you can’t predict which ones will be chosen; any combination could be chosen.
2. A random sample drawn from a population with a large cluster of points near the maximum is likely to contain at least one element near the maximum. If any of the elements have similar values, it appears that the chances of getting one of them in a random sample are high.
3. The variation of sample statistics from sample to sample is called sampling variability.
Example using Multiple Random Samples
You and a group of friends want to know how many of your school’s students prefer hip hop dance. Your school has a total of 840 students. Each member of the group surveys 20 students at random. The outcomes are given in the table below.
Use each sample to make an estimate for the number of students in your school who prefer hip hop dance.
Solution:
In the first sample, 13 out of 20, or 65% of the students choose hip hop dance.
So, you can estimate that 0.65 (840) = 546 students in the school prefer hip hop dance than tap dance, rock and roll, and swing dance.
So, the estimates are that 336, 378, 420, and 546 students prefer hip hop dance.
Mean in Statistics
Mean is the ratio of the sum of observations of a data divided by the number of observations. The formula is below.
\( \text{Mean X}= \frac{\text{Sum of observations}}{\text{Number of observations}}=\frac{\sum xi}{n} \)
1. A population characteristic is estimated by selecting a random sample of the population and computing the value of a statistic for the sample. A population mean, for example, can be estimated by taking a random sample of the population and calculating the sample mean.
2. The sample statistic (e.g., the sample mean) will have a different value depending on the random sample used. The variation in the values of the sample statistic from one sample to the next is referred to as sampling variability.
3.The population proportion will be approximately equal to the mean of the sample proportions.
Example on Estimating an Average of a Population
1. You want to know the mean number of hours students attend music classes each week. At each of six schools you randomly survey 10 students attending the music classes. Your results are shown.
A. Make an estimate for the average number of hours students spend in music classes each week using each sample. Describe how the estimates differed.
B. What is the average of each sample? Make a single estimate for the average number of hours students spend in music classes each week using all six samples.
Solution:
A.
Sample | a | b | c | d | e | f |
|---|---|---|---|---|---|---|
Mean | \(\frac{70}{10}=7\) | \(\frac{73}{10}=7\) | \(\frac{77}{10}=7\) | \(\frac{50}{10}=7\) | \(\frac{80}{10}=7\) | \(\frac{90}{10}=7\) |
Since, \( \text{mean}=\frac{\text{Sum of observations}}{\text{Number of observations}}=\frac{(\sum xi)}{n}\)
So, the six estimates for students with music classes are; 5, 7, 7.3, 7.7, 8, and 9 hours for each week.
As we know the range is the (highest observation – lowest observation).
The estimates have a range of 9 – 5 = 4 hours.
B. The mean of all sample data is \( \frac{440}{60}=7.3\) hours
So, you can estimate that students attend music classes 7.3 hours in a week.
2. The mean of the random samples 10,22,38,43,P and 27 is 25. Find the value of P?
Solution:
We know that, \( \text{mean}=\frac{\text{Sum of observations}}{\text{Number of observations}}=\frac{(\sum xi)}{n}\)
\( 25=\frac{(10+22+38+43+P+27)}{6} \)
\( 150=140+P \)
\( P=10 \)
3. Rainfall of a place in a week is 5cm, 6cm, 13cm, 4cm, 7cm, 9cm, 1.5cm. Find the average rainfall per day.
Solution:
The average rainfall per day is the arithmetic mean of the above observations.
The rainfall through a week from the given question are 5cm, 6cm, 13cm, 4cm, 7cm, 9cm, 1.5cm.
\( \text{mean}=\frac{\text{Sum of observations}}{\text{Number of observations}}=\frac{(\sum xi)}{n}\)
\( =\frac{X1+X2+X3+…..+Xn}{n}\)
Where X1 +X2 +X3 +…….Xn are n observations and X is their mean.
\( =\frac{5+6+13+4+7+9+15}{7}\)
\( =\frac{45.5}{7}\)
\( =6.5 \) cm.
Therefore, the average rainfall per day = 6.5cm.
Median in Statistics
Median is the middle observation of a certain data set. When it is arranged in an order (ascending/descending), it divides the data into two groups of equal strength, one group consisting of all values greater than the median and the other group comprising values less than the median.
When the data has ‘n’ number of observations and if ‘n’ is odd, median is \( (\frac{n+1}{2}) \) th observation.
When n is even, median is the average of the \( (\frac{n}{2}) \) th and \( (\frac{n}{2}+1) \) th observation.
Mode is the value of the observation which occurs most frequently, i.e., an observation with the maximum frequency is called the mode.
Examples on Median of the given Data
Find the median of scores 75, 21, 56, 36, 81, 05, 42?
Solution:
Number of observations = 7
As the number of observations(n) is an odd number, so
\( \text{Meadian}=(\frac{n+1}{2}) \) th observation.
\( =4 \) th observation
Therefore, the median is 36 as per the given scores in the question.
To choose a sample, simple random sampling requires the use of randomly generated numbers. It includes a sampling frame, a list, or a database of all members of a population at first. You can then generate a random number for each element, for example, in Excel, and take the first n samples you need.
Random sampling can be divided into four types:
1. A simple and direct random sampling.
2 Stratified random sampling.
3. Random sampling in clusters.
4. Random sampling in a systematic manner.