<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Gathering information about an entire population often costs too much or is virtually impossible. Instead, we use a sample of the population.&nbsp;<strong>A sample should have the same characteristics as the population it is representing.</strong>&nbsp;Most statisticians use various methods of random sampling in an attempt to achieve this goal. This section will describe a few of the most common methods. There are several different methods of&nbsp;<strong>random sampling</strong>. In each form of random sampling, each member of a population initially has an equal chance of being selected for the sample. Each method has pros and cons. The easiest method to describe is called a&nbsp;<strong>simple random sample</strong>. Any group of&nbsp;<em>n</em>&nbsp;individuals is equally likely to be chosen as any other group of&nbsp;<em>n</em>&nbsp;individuals if the simple random sampling technique is used. In other words, each sample of the same size has an equal chance of being selected.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Besides simple random sampling, there are other forms of sampling that involve a chance process for getting the sample.&nbsp;<strong>Other well-known random sampling methods are the stratified sample, the cluster sample, and the systematic sample.</strong></span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">To choose a&nbsp;<strong>stratified sample</strong>, divide the population into groups called strata and then take a&nbsp;<strong>proportionate</strong>&nbsp;number from each stratum. For example, you could stratify (group) your college population by department and then choose a proportionate simple random sample from each stratum (each department) to get a stratified random sample. To choose a simple random sample from each department, number each member of the first department, number each member of the second department, and do the same for the remaining departments. Then use simple random sampling to choose proportionate numbers from the first department and do the same for each of the remaining departments. Those numbers picked from the first department, picked from the second department, and so on represent the members who make up the stratified sample.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">To choose a&nbsp;<strong>cluster sample</strong>, divide the population into clusters (groups) and then randomly select some of the clusters. All the members from these clusters are in the cluster sample. For example, if you randomly sample four departments from your college population, the four departments make up the cluster sample. Divide your college faculty by department. The departments are the clusters. Number each department, and then choose four different numbers using simple random sampling. All members of the four departments with those numbers are the cluster sample.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">To choose a&nbsp;<strong>systematic sample</strong>, randomly select a starting point and take every&nbsp;<em>n</em><sup>th</sup>&nbsp;piece of data from a listing of the population. For example, suppose you have to do a phone survey. Your phone book contains 20,000 residence listings. You must choose 400 names for the sample. Number the population 1&ndash;20,000 and then use a simple random sample to pick a number that represents the first name in the sample. Then choose every fiftieth name thereafter until you have a total of 400 names (you might have to go back to the beginning of your phone list). Systematic sampling is frequently chosen because it is a simple method.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">A type of sampling that is non-random is convenience sampling.&nbsp;<strong>Convenience sampling</strong>&nbsp;involves using results that are readily available. For example, a computer software store conducts a marketing study by interviewing potential customers who happen to be in the store browsing through the available software. The results of convenience sampling may be very good in some cases and highly biased (favor certain outcomes) in others.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Sampling data should be done very carefully. Collecting data carelessly can have devastating results. Surveys mailed to households and then returned may be very biased (they may favor a certain group). It is better for the person conducting the survey to select the sample respondents.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">True random sampling is done&nbsp;<strong>with replacement</strong>. That is, once a member is picked, that member goes back into the population and thus may be chosen more than once. However for practical reasons, in most populations, simple random sampling is done&nbsp;<strong>without replacement</strong>. Surveys are typically done without replacement. That is, a member of the population may be chosen only once. Most samples are taken from large populations and the sample tends to be small in comparison to the population. Since this is the case, sampling without replacement is approximately the same as sampling with replacement because the chance of picking the same individual more than once with replacement is very low.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">In a college population of 10,000 people, suppose you want to pick a sample of 1,000 randomly for a survey.&nbsp;<strong>For any particular sample of 1,000</strong>, if you are sampling&nbsp;<strong>with replacement</strong>,</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">the chance of picking the first person is 1,000 out of 10,000 (0.1000);</span></span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">the chance of picking a different second person for this sample is 999 out of 10,000 (0.0999);</span></span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">the chance of picking the same person again is 1 out of 10,000 (very low).</span></span></span></p>
	</li>
</ul>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">If you are sampling&nbsp;<strong>without replacement</strong>,</span></span></span></p>

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">the chance of picking the first person for any particular sample is 1000 out of 10,000 (0.1000);</span></span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">the chance of picking a different second person is 999 out of 9,999 (0.0999);</span></span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">you do not replace the first person before picking the next person.</span></span></span></p>
	</li>
</ul>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Compare the fractions 999/10,000 and 999/9,999. For accuracy, carry the decimal answers to four decimal places. To four decimal places, these numbers are equivalent (0.0999).</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Sampling without replacement instead of sampling with replacement becomes a mathematical issue only when the population is small. For example, if the population is 25 people, the sample is ten, and you are sampling&nbsp;<strong>with replacement for any particular sample</strong>, then the chance of picking the first person is ten out of 25, and the chance of picking a different second person is nine out of 25 (you replace the first person).</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">If you sample&nbsp;<strong>without replacement</strong>, then the chance of picking the first person is ten out of 25, and then the chance of picking the second person (who is different) is nine out of 24 (you do not replace the first person).</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">Compare the fractions 9/25 and 9/24. To four decimal places, 9/25 = 0.3600 and 9/24 = 0.3750. To four decimal places, these numbers are not equivalent.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">When you analyze data, it is important to be aware of&nbsp;<strong>sampling errors</strong>&nbsp;and nonsampling errors. The actual process of sampling causes sampling errors. For example, the sample may not be large enough. Factors not related to the sampling process cause&nbsp;<strong>nonsampling errors</strong>. A defective counting device can cause a nonsampling error.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">In reality, a sample will never be exactly representative of the population so there will always be some sampling error. As a rule, the larger the sample, the smaller the sampling error.</span></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="background-color:white"><span style="color:#424242">In statistics,&nbsp;<strong>a sampling bias</strong>&nbsp;is created when a sample is collected from a population and some members of the population are not as likely to be chosen as others (remember, each member of the population should have an equally likely chance of being chosen). When a sampling bias happens, there can be incorrect conclusions drawn about the population that is being studied.</span></span></span></p>

<h4 style="text-align:justify">&nbsp;</h4>

<h4 style="text-align:justify"><strong><span style="font-size:14px"><span style="color:#2e74b5"><em><span style="color:#333333">Critical Evaluation</span></em></span></span></strong></h4>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#424242">We need to evaluate the statistical studies we read about critically and analyze them before accepting the results of the studies. Common problems to be aware of include</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Problems with samples: A sample must be representative of the population. A sample that is not representative of the population is biased. Biased samples that are not representative of the population give results that are inaccurate and not valid.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Self-selected samples: Responses only by people who choose to respond, such as call-in surveys, are often unreliable.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Sample size issues: Samples that are too small may be unreliable. Larger samples are better, if possible. In some situations, having small samples is unavoidable and can still be used to draw conclusions. Examples: crash testing cars or medical testing for rare conditions</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Undue influence: &nbsp;collecting data or asking questions in a way that influences the response</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Non-response or refusal of subject to participate: &nbsp;The collected responses may no longer be representative of the population. &nbsp;Often, people with strong positive or negative opinions may answer surveys, which can affect the results.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Causality: A relationship between two variables does not mean that one causes the other to occur. They may be related (correlated) because of their relationship through a different variable.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Self-funded or self-interest studies: A study performed by a person or organization in order to support their claim. Is the study impartial? Read the study carefully to evaluate the work. Do not automatically assume that the study is good, but do not automatically assume the study is bad either. Evaluate it on its merits and the work done.</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Misleading use of data: improperly displayed graphs, incomplete data, or lack of context</span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:#424242">Confounding: &nbsp;When the effects of multiple factors on a response cannot be separated. &nbsp;Confounding makes it difficult or impossible to draw valid conclusions about the effect of each factor.</span></span></p>
	</li>
</ul>