The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter M is often used to represent the median. The next example illustrates the location of the median and the value of the median. To find the median, M , first use the formula for the location. The location is:. Starting at the smallest value, the median is located between the 20th and 21st values the two 24s :. Which is the better measure of the "center," the mean or the median?
The median is a better measure of the "center" than the mean because 49 of the values are 30, and one is 5,, The 5,, is an outlier. The 30, gives us a better sense of the middle of the data.
Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal. Five real estate exam scores are , , , , The data set is bimodal because the scores and each occur twice. When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program.
The mode might indicate that most people lose two pounds the first week, making the program less appealing. Statistical software will easily calculate the mean, the median, and the mode.
Some graphing calculators can also make these calculations. In the real world, people make these calculations using software. This is discussed in more detail in The Central Limit Theorem.
The formula for the mean is located in the Summary of Formulas section course. You can think of a sampling distribution as a relative frequency distribution with a great many samples. See Sampling and Data for a review of relative frequency. Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below. If you let the number of samples get very large say, million or more , the relative frequency table becomes a relative frequency distribution.
A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. This data set produces the histogram shown below.
Each interval has width one and each value is located in the middle of an interval. The histogram displays a symmetrical distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other.
The mean, the median, and the mode are each 7 for these data. In a perfectly symmetrical distribution, the mean and the median are the same. This example has one mode unimodal and the mode is the same as the mean and median.
In a symmetrical distribution that has two modes bimodal , the two modes would be different from the mean and median. The right-hand side seems "chopped off" compared to the left side. The shape distribution is called skewed to the left because it is pulled out to the left. The mean is 6. Notice that the mean is less than the median and they are both less than the mode.
The mean and the median both reflect the skewing but the mean more so. It is skewed to the right. The mean is 7. Of the three statistics, the mean is the largest, while the mode is the smallest.
Again, the mean reflects the skewing the most. To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.
Skewness and symmetry become important when we discuss probability distributions in later chapters. Descriptive Statistics: Measuring the Spread of Data explains standard deviation as a measure of variation in data and is part of the collection col written by Barbara Illowsky and Susan Dean.
Roberta Bloom made contributions that helped to clarify the standard deviation and the variance. An important characteristic of any set of data is the variation in the data.
In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation.
The standard deviation is a number that measures how far data values are from their mean. The standard deviation is always positive or 0. The standard deviation is small when the data are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying waiting times at the checkout line for customers at supermarket A and supermarket B; the average wait time at both markets is 5 minutes. At market A, the standard deviation for the waiting time is 2 minutes; at market B the standard deviation for the waiting time is 4 minutes. Because market B has a higher standard deviation, we know that there is more variation in the waiting times at market B.
Overall, wait times at market B are more spread out from the average; wait times at market A are more concentrated near the average.
Suppose that Rosa and Binh both shop at Market A. Rosa waits for 7 minutes and Binh waits for 1 minute at the checkout counter. At market A, the mean wait time is 5 minutes and the standard deviation is 2 minutes.
The standard deviation can be used to determine whether a data value is close to or far from the mean. Rosa's wait time of 7 minutes is 2 minutes longer than the average of 5 minutes. Rosa's wait time of 7 minutes is one standard deviation above the average of 5 minutes.
Binh's wait time of 1 minute is 4 minutes less than the average of 5 minutes. Binh's wait time of 1 minute is two standard deviations below the average of 5 minutes.
A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if it is more than 2 standard deviations away is more of an approximate "rule of thumb" than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is further away than 2 standard deviations. We will learn more about this in later chapters.
The number line may help you understand standard deviation. If we were to put 5 and 7 on a number line, 7 is to the right of 5.
If x is a number, then the difference " x - mean" is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. For sample data, in symbols a deviation is x —. The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar, but not identical. Therefore the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample.
To calculate the standard deviation, we need to calculate the variance first. The symbol s 2 represents the sample variance; the sample standard deviation s is the square root of the sample variance.
You can think of the standard deviation as a special average of the deviations. If the numbers come from a census of the entire population and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by N , the number of items in the population.
If the data are from a sample rather than a population, when we calculate the average of the squared deviations, we divide by n-1 , one less than the number of items in the sample. You can see that in the formulas below. For the population standard deviation, the denominator is N , the number of items in the population. In these formulas, f represents the frequency with which a value appears. For example, if a value appears once, f is 1. If a value appears three times in the data set or population, f is 3.
The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data. How much the statistic varies from one sample to another is known as the sampling variability of a statistic. You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean.
You will cover the standard error of the mean in The Central Limit Theorem not now. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean. In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students.
The ages are rounded to the nearest half year:. The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating s. The sample variance, s 2 , is equal to the sum of the last column 9. The sample standard deviation s is equal to the square root of the sample variance:. Typically, you do the calculation for the standard deviation on your calculator or computer.
The intermediate results are not rounded. This is done for accuracy. Verify the mean and standard deviation calculated above on your calculator or computer. Enter data into the list editor.
If necessary, clear the lists by arrowing up into the name. Put the data values 9, 9. Use the arrow keys to move around. Press VarStats and enter L1 2nd 1 , L2 2nd 2. Do not forget the comma. Find the value that is 1 standard deviation above the mean. Find the value that is two standard deviations below the mean. Find the values that are 1. The deviations show how spread out the data are about the mean.
The data value The deviations 0. A positive deviation occurs when the data value is greater than the mean. A negative deviation occurs when the data value is less than the mean; the deviation is If you add the deviations, the sum is always zero. So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive.
The variance, then, is the average squared deviation. The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data. For the sample variance, we divide by the sample size minus one n Why not divide by n? The answer has to do with the population variance. The sample variance is an estimate of the population variance. Based on the theoretical mathematics that lies behind these calculations, dividing by n-1 gives a better estimate of the population variance.
Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number which measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic. When the standard deviation is 0, there is no spread; that is, the all the data values are equal to each other.
The standard deviation is small when the data are all concentrated close to the mean, and is larger when the data values show more variation from the mean. The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better "feel" for the deviations and the standard deviation.
You will find that in symmetrical distributions, the standard deviation can be very helpful but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing, always graph your data. Use the following data first exam scores from Susan Dean's spring pre-calculus class:.
The long left whisker in the box plot is reflected in the left side of the histogram. The histogram, box plot, and chart all reflect this. There are a substantial number of A and B grades 80s, 90s, and The histogram clearly shows this. The standard deviation is useful when comparing data values that come from different data sets. If the data sets have different means and standard deviations, it can be misleading to compare the data values directly.
In symbols, the formulas become:. Two students, John and Ali, from different high schools, wanted to find out who had the highest G. Which student had the highest G. Pay careful attention to signs when comparing and interpreting the answer. For John,. For Ali,. John has the better G. The following lists give a few facts that provide a little more insight into what the standard deviation tells us about the distribution of the data.
It is important to note that this rule only applies when the shape of the distribution of the data is mound-shaped and symmetric. We will learn more about this when studying the "Normal" or "Gaussian" probability distribution in later chapters. This module provides students with opportunities to apply concepts related to descriptive statistics. Students are asked to take a set of sample data and calculate a series of statistical values for that data.
Sixty-five randomly selected car salespersons were asked the number of cars they generally sell in one week. Fourteen people answered that they generally sell three cars; nineteen generally sell four cars; twelve generally sell five cars; nine generally sell six cars; eleven generally sell seven cars.
What is the difference between relative frequency and frequency for each data value? What is the difference between cumulative relative frequency and relative frequency for each data value? Determine appropriate minimum and maximum x and y values and the scaling. Sketch the histogram below. Label the horizontal and vertical axes with words. Include numerical scaling. Looking at your box plot, does it appear that the data are concentrated together, spread out evenly, or concentrated in some areas, but not in others?
How can you tell? The population parameters below describe the full-time equivalent number of students FTES each year at Lake Tahoe Community College from through A sample of 11 years is taken. About how many are expected to have a FTES of or above? Explain how you determined your answer. What is the IQR?
What does the IQR represent? The data are reported here. Calculate the mean, median, standard deviation, first quartile, the third quartile and the IQR. Round to one decimal place. Why do you suppose the IQRs are so different? Descriptive Statistics: Homework is part of the collection col written by Barbara Illowsky and Susan Dean and provides homework questions related to lessons about descriptive statistics. Twenty-five randomly selected students were asked the number of movies they watched the previous week.
The results are as follows:. The median age for U. Forty randomly selected students were asked the number of pairs of sneakers they owned. The results are below.
Also, include left endpoint, but not the right endpoint. Source: Time magazine; survey by Yankelovich Partners, Inc. Why or why not? Following are the published weights in pounds of all of the team members of the San Francisco 49ers from a previous year Source: San Jose Mercury News.
An elementary school class ran 1 mile with a mean of 11 minutes and a standard deviation of 3 minutes. Rachel, a student in the class, ran 1 mile in 8 minutes. A junior high school class ran 1 mile with a mean of 9 minutes and a standard deviation of 2 minutes. Kenji, a student in the class, ran 1 mile in 8.
A high school class ran 1 mile with a mean of 7 minutes and a standard deviation of 4 minutes. Nedda, a student in the class, ran 1 mile in 8 minutes. In a survey of 20 year olds in China, Germany and America, people were asked the number of foreign countries they had visited in their lifetime. The following box plots display the results. One hundred teachers attended a seminar on mathematical problem solving. The attitudes of a representative sample of 12 of the teachers were measured before and after the seminar.
A positive number for change in attitude indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows:. Three students were applying to the same graduate school.
They came from schools with different grading systems. Which student had the best G. Their ages are as follows. Source: West magazine. Suppose that three book publishers were interested in the number of fiction paperbacks adult consumers purchase per month.
Each publisher conducted a survey. In the survey, each asked adult consumers the number of fiction paperbacks they had purchased the previous month. Often, cruise ships conduct all on-board transactions, with the exception of gambling, on a cashless basis. At the end of the cruise, guests pay one bill that covers all on-board transactions. Suppose that 60 single travelers and 70 couples were surveyed as to their on-board bills for a seven-day cruise from Los Angeles to the Mexican Riviera.
Below is a summary of the bills for each group. Refer to the following histograms and box plot. Hiltz The through years of operation, examinations were seldom collaboration discussed in this paper is not collaboration conducted on-line. Distance learning students usually within groups, but among the class as a whole.
Each have to commute to designated campuses to take exams student had to contribute and interact with other students or have an approved proctor. Before we introduce the throughout the process, and the exam was conducted evaluation of an on-line examination procedure at NJIT, through cooperation with the whole class. Ebel and Frisbie learning process that emphasizes group or cooperative point out that the main purposes of a test are to measure efforts among faculty and students Hiltz It student achievement and motivate and direct student stresses active participation and interaction for both learning; and also the process of taking an exam and students and instructors.
In collaborative learning, discussing its scoring should be a richly rewarding instruction is learner-centered rather than teacher- learning experience. In addition, the process of centered, and knowledge is viewed as a social construct, constructing tests should cause instructors to think facilitated by peer interaction, evaluation and cooperation. Later, Ph. Though different from the and lastly, the instructor provides a final grade.
Mason and Woit Classroom. The instructor created three conferences for highlighted the potential advantages of on-line the exam. Conferences are separate discussion areas examinations by encouraging students to attain practical where students can post and read the comments others skills, and reducing level of cheating and copying.
The post. Each conference has a number and a name. The analysis by Kumar indicated that there was a good functions of the three conferences were as follows: 1 correlation between written test and on-line test scores.
Students posted their questions, answers, grading justifications, and Ph. The main procedure of the midterm took place over a for both masters and Ph. The course Survey and Questionnaire covers one topic for each lecture, with related textbook chapters and articles as instruction materials.
There are To evaluate the collaborative on-line examination two sections of the course: a Face-to-Face FTF section process, a survey was conducted within a couple of weeks where students meet once a week; and a Distance- after the main examination process finished. The Learning DL section where students watch video tapes questionnaire was distributed in the classroom to FTF of the lectures. Students in both sections are required to students and also was put on the web site of the instructor participate in the on-line discussion in Virtual for DL students.
As a result, the percentage of papers that are entirely 'home grown' is falling. In emerging economies, by contrast, domestic output is rapidly expanding see 'Strength in numbers'. These shifts stand to change the dynamics of how nations invest in science, who owns its outputs and who best exploits them. In Western Europe in , domestic share of output was about five papers in every six published by each country, but fell to half of total output in Nonetheless, it remains less internationally collaborative than Western Europe.
In emerging economies in , the domestic share of output was similar to that of the established economies. The total volume of papers from these four countries has increased fold — from fewer than 15, papers annually in to more than , papers now.
Despite its European location, Poland's balance of domestic and collaborative papers is closer to this group see Supplementary Information.
For established economies, total national research output has more than doubled over the past 30 years. After the mids, the domestic research output of the United Kingdom 47, papers per year , Germany 45, papers and France 30, papers levelled off while international collaboration in these countries increased more than ten-fold.
What fuelled this growth in collaboration? Much of it has come from bilateral partnerships rather than multinational programmes see Supplementary Information. Switzerland is an interesting outlier. For a small country, its research output has seen above-average growth and a high average citation impact. Today, more than two-thirds of its publications have a foreign co-author. A large part of this comes from the country's hosting of multinational programmes, such as CERN Europe's particle-physics lab and the World Health Organization, both in Geneva.
Citation impact is typically greater when research groups collaborate, and the benefit strengthens when co-authorship is international 4 , 5 , 6. I calculated the citation impact of each UK and US paper published in and then for by normalizing citation count to take into account the time elapsed since year of publication and the subject area of the journal.
For each of these years, I then calculated an average normalized citation impact where 1. For both countries, papers that are internationally collaborative are cited relatively more often than purely domestic papers. A closer look at the United Kingdom reveals the drivers of some of these changes.
My analysis shows that this impact up from 1. By comparison, the UK domestic research base, which BIS has never analysed separately, has changed much less in quantity and quality. In , the country published 46, domestic papers with an average impact of 1. And in , the number of UK papers with no overseas co-author actually fell below the count of internationally collaborative papers.
The benefits of collaboration are systematic. I found that UK universities and colleges published more than research articles and reviews in journals indexed on Thomson Reuters Web of Science in — Average impact in this group rarely gets much above 1. This category includes the five or six UK universities that are generally recognized as standing ahead of others, including the universities of Oxford and Cambridge.
This same handful of elite institutions receives about one-third of all available core research income from the UK higher-education funding councils, more than one-third of the total available research grant and contract income and one-third of all research-council studentships.
First-rate researchers are, and always were, thinly distributed. In the past they could collaborate only intermittently. Keyword search:. Full Text. Citation Illowsky, B. This record was imported from OpenPraxis on September 12, Higher Education. Babson Survey Research Group, October Retrieved from www.
Educause Learning Initiative Brief, January Personal communication, copy in possession of the author. Hewlett Open educational resources. Cost-savings achieved in two semesters through the adoption of open educational resources.
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