Show A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (or mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose. When a statistic is used for estimating a population parameter, the statistic is called an estimator. A population parameter is any characteristic of a population under study, but when it is not feasible to directly measure the value of a population parameter, statistical methods are used to infer the likely value of the parameter on the basis of a statistic computed from a sample taken from the population. For example, the sample mean is an unbiased estimator of the population mean. This means that the expected value of the sample mean equals the true population mean.[1] A descriptive statistic is used to summarize the sample data. A test statistic is used in statistical hypothesis testing. Note that a single statistic can be used for multiple purposes – for example the sample mean can be used to estimate the population mean, to describe a sample data set, or to test a hypothesis. ExamplesSome examples of statistics are:
In this case, "52%" is a statistic, namely the percentage of Republicans in the survey sample who believe in global warming. The population is the set of all Republicans in the United States, and the population parameter being estimated is the percentage of all Republicans in the United States, not just those surveyed, who believe in global warming.
In this example, "5.6 days" is a statistic, namely the mean length of stay for our sample of 20 hotel guests. The population is the set of all guests of this hotel, and the population parameter being estimated is the mean length of stay for all guests.[2] Note that whether the estimator is unbiased in this case depends upon the sample selection process; see the inspection paradox. There are a variety of functions that are used to calculate statistics. Some include:
PropertiesObservabilityStatisticians often contemplate a parameterized family of probability distributions, any member of which could be the distribution of some measurable aspect of each member of a population, from which a sample is drawn randomly. For example, the parameter may be the average height of 25-year-old men in North America. The height of the members of a sample of 100 such men are measured; the average of those 100 numbers is a statistic. The average of the heights of all members of the population is not a statistic unless that has somehow also been ascertained (such as by measuring every member of the population). The average height that would be calculated using all of the individual heights of all 25-year-old North American men is a parameter, and not a statistic. Statistical propertiesImportant potential properties of statistics include completeness, consistency, sufficiency, unbiasedness, minimum mean square error, low variance, robustness, and computational convenience. Information of a statisticInformation of a statistic on model parameters can be defined in several ways. The most common is the Fisher information, which is defined on the statistic model induced by the statistic. Kullback information measure can also be used. See also
 Look up statistic in Wiktionary, the free dictionary.
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1. A Normal density curve (or distribution) has which of the following properties? A. It has a peak centered above its mean B. It is skewed C. The standard deviation is always equal to one D. All of the above 2. Assume X is a normally distributed random variable with a known population mean of 240 and a known population standard deviation 30. What can we conclude? A. Approximately 99.7% of the values of X are between 150 and 330. B. Approximately 99.7% of the values of X are between 210 and 270. C. Approximately 99.7% of the values of X are between 180 and 300. D. Approximately 95% of the values of X are between 150 and 330. 3. Scores on the national Pharmacy D. certification exam are approximately normally distributed with a mean of 575 points and a standard deviation of 42 points. If one person is selected at random, what is the probability that person will have score less than 600 points? A. 0.6000 B. 0.2743 C. 0.7257 D. 0.9583
The science of statistics deals with the collection, analysis, interpretation, and presentation of data. We see and use data in our everyday lives. Learning Outcomes
In your classroom, try this exercise. Have class members write down the average time (in hours, to the nearest half-hour) they sleep per night. Your instructor will record the data. Then create a simple graph (called a dot plot) of the data. A dot plot consists of a number line and dots (or points) positioned above the number line. For example, consider the following data: The dot plot for this data would be as follows: Does your dot plot look the same as or different from the example? Why? If you did the same example in an English class with the same number of students, do you think the results would be the same? Why or why not? Where do your data appear to cluster? How might you interpret the clustering? The questions above ask you to analyze and interpret your data. With this example, you have begun your study of statistics. In this course, you will learn how to organize and summarize data. Organizing and summarizing data is called descriptive statistics. Two ways to summarize data are by graphing and by using numbers (for example, finding an average). After you have studied probability and probability distributions, you will use formal methods for drawing conclusions from “good” data. The formal methods are called inferential statistics. Statistical inference uses probability to determine how confident we can be that our conclusions are correct. Effective interpretation of data (inference) is based on good procedures for producing data and thoughtful examination of the data. You will encounter what will seem to be too many mathematical formulas for interpreting data. The goal of statistics is not to perform numerous calculations using the formulas, but to gain an understanding of your data. The calculations can be done using a calculator or a computer. The understanding must come from you. If you can thoroughly grasp the basics of statistics, you can be more confident in the decisions you make in life. ProbabilityProbability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. For example, if you toss a fair coin four times, the outcomes may not be two heads and two tails. However, if you toss the same coin [latex]4,000[/latex] times, the outcomes will be close to half heads and half tails. The expected theoretical probability of heads in any one toss is [latex]12[/latex] or [latex]0.5[/latex]. Even though the outcomes of a few repetitions are uncertain, there is a regular pattern of outcomes when there are many repetitions. After reading about the English statistician Karl Pearson who tossed a coin [latex]24,000[/latex] times with a result of [latex]12,012[/latex] heads, one of the authors tossed a coin [latex]2,000[/latex] times. The results were [latex]996[/latex] heads. The fraction [latex]\displaystyle\frac{{996}}{{2,000}}[/latex] is equal to [latex]0.498[/latex] which is very close to [latex]0.5[/latex], the expected probability. The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client’s investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data. Key TermsIn statistics, we generally want to study a population. You can think of a population as a collection of persons, things, or objects under study. To study the population, we select a sample. The idea of sampling is to select a portion (or subset) of the larger population and study that portion (the sample) to gain information about the population. Data are the result of sampling from a population. 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’ grade point averages. In presidential elections, opinion poll samples of [latex]1,000[/latex]–[latex]2,000[/latex] 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 [latex]16[/latex] ounce can contains [latex]16[/latex] ounces of carbonated drink. From the sample data, we can calculate a statistic. A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all 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. The statistic is an estimate of a population parameter. A parameter is a number that is a property of the population. Since we 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. One of the main concerns in the field of statistics is how accurately a statistic estimates a parameter. The accuracy really depends on how well the sample represents the population. The sample must contain the characteristics of the population in order to be a representative sample. 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. A variable, notated by capital letters such as [latex]X[/latex] and [latex]Y[/latex], is a characteristic of interest for each person or thing in a population. Variables may be numerical or categorical. Numerical variables take on values with equal units such as weight in pounds and time in hours. Categorical variables place the person or thing into a category. If we let [latex]X[/latex] equal the number of points earned by one math student at the end of a term, then [latex]X[/latex] is a numerical variable. If we let [latex]Y[/latex] be a person’s party affiliation, then some examples of [latex]Y[/latex] include Republican, Democrat, and Independent. [latex]Y[/latex] is a categorical variable. We could do some math with values of [latex]X[/latex] (calculate the average number of points earned, for example), but it makes no sense to do math with values of [latex]Y[/latex] (calculating an average party affiliation makes no sense). Data are the actual values of the variable. They may be numbers or they may be words. Datum is a single value. Two words that come up often in statistics are mean and proportion. If you were to take three exams in your math classes and obtain scores of [latex]86[/latex], [latex]75[/latex], and [latex]92[/latex], you would calculate your mean score by adding the three exam scores and dividing by three (your mean score would be [latex]84.3[/latex] to one decimal place). If, in your math class, there are [latex]40[/latex] students and [latex]22[/latex] are men and [latex]18[/latex] are women, then the proportion of men students is [latex]\displaystyle\frac{{22}}{{40}}[/latex] and the proportion of women students is [latex]\displaystyle\frac{{18}}{{40}}[/latex]. Mean and proportion are discussed in more detail in later chapters.
The words “mean” and “average” are often used interchangeably. The substitution of one word for the other is common practice. The technical term is “arithmetic mean,” and “average” is technically a center location. However, in practice among non-statisticians, “average” is commonly accepted for “arithmetic mean.”
Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money first year college students spend at ABC College on school supplies that do not include books. We randomly survey [latex]100[/latex] first year students at the college. Three of those students spent [latex]$150[/latex], [latex]$200[/latex], and [latex]$225[/latex], respectively.
Determine what the key terms refer to in the following study. We want to know the average (mean) amount of money spent on school uniforms each year by families with children at Knoll Academy. We randomly survey [latex]100[/latex] families with children in the school. Three of the families spent [latex]$65[/latex], [latex]$75[/latex], and [latex]$95[/latex], respectively.
Determine what the key terms refer to in the following study. 1._____ Population 2._____ Statistic 3._____ Parameter 4._____ Sample 5._____ Variable 6._____ Data a) all students who attended the college last year b) the cumulative GPA of one student who graduated from the college last year c) [latex]3.65[/latex], [latex]2.80[/latex], [latex]1.50[/latex], [latex]3.90[/latex] d) a group of students who graduated from the college last year, randomly selected e) the average cumulative GPA of students who graduated from the college last year f) all students who graduated from the college last year g) the average cumulative GPA of students in the study who graduated from the college last year
Determine what the key terms refer to in the following study.
Cars with dummies in the front seats were crashed into a wall at a speed of [latex]35[/latex] miles per hour. We want to know the proportion of dummies in the driver’s seat that would have had head injuries, if they had been actual drivers. We start with a simple random sample of [latex]75[/latex] cars.
Determine what the key terms refer to in the following study. An insurance company would like to determine the proportion of all medical doctors who have been involved in one or more malpractice lawsuits. The company selects [latex]500[/latex] doctors at random from a professional directory and determines the number in the sample who have been involved in a malpractice lawsuit.
Do the following exercise collaboratively with up to four people per group. Find a population, a sample, the parameter, the statistic, a variable, and data for the following study: You want to determine the average (mean) number of glasses of milk college students drink per day. Suppose yesterday, in your English class, you asked five students how many glasses of milk they drank the day before. The answers were [latex]1[/latex], [latex]0[/latex], [latex]1[/latex], [latex]3[/latex], and [latex]4[/latex] glasses of milk. Watch the following video for a brief introduction to statistics. Concept ReviewThe mathematical theory of statistics is easier to learn when you know the language. This module presents important terms that will be used throughout the text. GlossaryAverage also called mean; a number that describes the central tendency of the data Categorical Variable variables that take on values that are names or labels Data a set of observations (a set of possible outcomes); most data can be put into two groups: qualitative (an attribute whose value is indicated by a label) or quantitative (an attribute whose value is indicated by a number). Quantitative data can be separated into two subgroups: discrete and continuous. Data is discrete if it is the result of counting (such as the number of students of a given ethnic group in a class or the number of books on a shelf). Data is continuous if it is the result of measuring (such as distance traveled or weight of luggage) Numerical Variable variables that take on values that are indicated by numbers Parameter a number that is used to represent a population characteristic and that generally cannot be determined easily Population all individuals, objects, or measurements whose properties are being studied Probability a number between zero and one, inclusive, that gives the likelihood that a specific event will occur Proportion the number of successes divided by the total number in the sample Representative Sample a subset of the population that has the same characteristics as the population Sample a subset of the population studied Statistic a numerical characteristic of the sample; a statistic estimates the corresponding population parameter. Variable a characteristic of interest for each person or object in a populationReferencesThe Data and Story Library, http://lib.stat.cmu.edu/DASL/Stories/CrashTestDummies.html (accessed May 1, 2013). |
What is the term in statistics for any characteristic or attribute of a person place or thing?
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