<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">In statistics, we generally want to study a </span><strong><span style="color:black">population</span></strong><span style="color:black">. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a </span><strong><span style="color:black">sample</span></strong><span style="color:black">. The idea of </span><strong><span style="color:black">sampling </span></strong><span style="color:black">is to select a portion, or subset, of the larger population and study that portion&mdash;the sample&mdash;to gain information about the population. Data are the result of sampling from a population.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Because it takes a lot of time and money to examine an entire population, sampling is a very practical technique. If you wished to compute the overall grade point average at your school, it would make sense to select a sample of students who attend the school. The data collected from the sample would be the students&#39; grade point averages. In presidential elections, opinion poll samples of 1,000&ndash;2,000 people are taken. The opinion poll is supposed to represent the views of the people in the entire country. Manufacturers of canned carbonated drinks take samples to determine if a 16-ml can contain 16 ml of carbonated drink.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">From the sample data, we can calculate a statistic. A </span><strong><span style="color:black">statistic </span></strong><span style="color:black">is a number that represents a property of the sample. For example, if we consider one math class as a sample of the population of </span><em><span style="color:black">all </span></em><span style="color:black">math classes, then the average number of points earned by students in that one math class at the end of the term is an example of a statistic. Since we do not have the data for all math classes, that statistic is our best estimate of the average for the entire population of math classes. If we happen to have data for </span><em><span style="color:black">all </span></em><span style="color:black">math classes, we can find the population parameter. A </span><strong><span style="color:black">parameter </span></strong><span style="color:black">is a numerical characteristic of the whole population that can be estimated by a statistic. Since we&nbsp;considered all math classes to be the population, then the average number of points earned per student over all the math classes is an example of a parameter.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. In order to have an accurate sample, it must contain the characteristics of the population in order to be a </span><strong><span style="color:black">representative sample</span></strong><span style="color:black">. We are interested in both the sample statistic and the population parameter in inferential statistics. In a later chapter, we will use the sample statistic to test the validity of the established population parameter.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">A </span><strong><span style="color:black">variable</span></strong><span style="color:black">, usually notated by capital letters such as </span><em><span style="color:black">X </span></em><span style="color:black">and </span><em><span style="color:black">Y</span></em><span style="color:black">, is a characteristic or measurement that can be determined for each member of a population. Variables may describe values like weight in pounds or favorite subject in school. </span><strong><span style="color:black">Numerical variables </span></strong><span style="color:black">take on values with equal units such as weight in pounds and time in hours. </span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:black">Categorical variables </span></strong><span style="color:black">place the person or thing into a category. If we let </span><em><span style="color:black">X </span></em><span style="color:black">equal the number of points earned by one math student at the end of a term, then </span><em><span style="color:black">X </span></em><span style="color:black">is a numerical variable. If we let </span><em><span style="color:black">Y </span></em><span style="color:black">be a person&#39;s party affiliation, then some examples of </span><em><span style="color:black">Y </span></em><span style="color:black">include BJP, INC, AAP, NCP, JD. </span><em><span style="color:black">Y </span></em><span style="color:black">is a categorical variable. </span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">We could do some math with values of </span><em><span style="color:black">X</span></em><span style="color:black">&mdash;calculate the average number of points earned, for example&mdash;but it makes no sense to do math with values of </span><em><span style="color:black">Y</span></em><span style="color:black">&mdash;calculating an average party affiliation makes no sense.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><strong><span style="color:black">Data </span></strong><span style="color:black">are the actual values of the variable. They may be numbers or they may be words. </span><strong><span style="color:black">Datum </span></strong><span style="color:black">is a single value.</span></span></p>

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<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Two words that come up often in statistics are </span><strong><span style="color:black">mean </span></strong><span style="color:black">and </span><strong><span style="color:black">proportion</span></strong><span style="color:black">. If you were to take three exams in your math classes and obtain scores of 86, 75, and 92, you would calculate your mean score by adding the three exam scores and dividing by three. Your mean score would be 84.3 to one decimal place. If, in your math class, there are 40 students and 22 are </span>males and 18 females, then the proportion of men students is <span class="math-tex">\(\frac{22}{40}\)</span>&nbsp;and the proportion of women students is&nbsp;<span class="math-tex">\(\frac{18}{40}\)</span>. Mean and proportion are discussed in more detail in later chapters.</span></p>

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<p style="margin-left:-1px; text-align:justify"><span style="font-size:14px"><strong>NOTE:</strong></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The words </span><em><span style="color:black">mean </span></em><span style="color:black">and </span><em><span style="color:black">average </span></em><span style="color:black">are often used interchangeably. In this book, we use the term </span><em><span style="color:black">arithmetic mean </span></em><span style="color:black">for mean. The substitution of one word for the other is common practice. The technical term is &ldquo;arithmetic mean,&rdquo; and &ldquo;average&rdquo; is technically a center location. However in practice among non-statisticians, &ldquo;average&rdquo; is commonly accepted for &ldquo;arithmetical mean&rdquo;</span></span></p>

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