2.6: Measures of the Center of the Data

Measures of center generally tell us about the middle, or center, of a distribution. They are the mean, the median, and the mode. Each plays a useful role in Statistics. The Mean The mean, or arithmetic average, is calculated by adding all the data values and dividing by the number of values. Symbolically, this is expressed as, where is the statistical symbol for the mean, is the sum of the data values, and n . Sep 20, · Now, the mean, sometimes called the arithmetic mean, is the average or the expected value that measures the central value of a data set. It is found by adding all of the values in the data set and dividing it by the number of values.

The median what is the muscle behind your knee cap generally measyre better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers.

The mean is the most common measure of the center. The substitution of one word for the other is common practice.

what is an unconstitutional law each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. The location of the median and the value of the median are not the same.

The next example illustrates the location of the median and the value of the median. AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows smallest to largest :. The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest. Calculate the mean and median. Another measure of the center is the mode. The mode is the most frequent value.

There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest.

A data set with two modes is mwasure bimodal. 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 oof lose two pounds the first week, making the program less kean. The mode can be calculated for qualitative data as well as for quantitative data.

For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red. 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. Consider the annual earnings of workers at a factory. Watch the following video from Khan Academy on finding the measute, median and mode of a set of data. This or discussed in more detail later in the text. You can think of a sampling distribution as a how to edit a.dat file frequency distribution with a great many samples.

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 whzt very large say, million or morethe 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. When only grouped data is available, you do not know the individual data values we only know intervals and interval frequencies ; therefore, you cannot compute an exact mean for the data set.

What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies.

We simply need to modify the definition to fit within the restrictions of a frequency table. Since we do not know the individual data values we can instead find the midpoint of each interval. Find the best estimate of the class mean. Maris conducted a study on the effect foes playing video games has on memory recall. As part of her study, she compiled the following data:. Find the midpoint voes each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.

The midpoints are [latex]1. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occurring datum or data in your data set. The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges which lack specific values, the mean may seem impossible to calculate.

However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval. Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set. Skip to main content. Module 2: Descriptive Statistics. Search for:. Measures of the Center of the Data Learning Outcomes Recognize, describe, and calculate the measures of the center of data: mean, median, and mode.

Try It The following data show the number of months patients typically wait on a transplant list before getting surgery. Note The mode can be calculated for qualitative data as well as for quantitative data. Grade Interval Number of Students [latex]50— Try It Maris conducted a study on the effect that playing how to cite circuit court cases games has on raisins are good for what recall.

Show Solution Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers The midpoints are [latex]1. Licenses and Attributions. CC licensed content, Shared previously.

Sampling Distributions and Statistic of a Sampling Distribution

Oct 28, · Contributors and Attributions. The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by Estimated Reading Time: 6 mins. The two most widely used measures of the “center” of the data are the mean (average) and the median. To calculate the mean weight of 50 50 people, add the 50 50 weights together and divide by 50 To find the median weight of the 50 50 people, order the data . Choose 1 answer: It is the middle point in the set of scores. That is, half the team scored higher than the mean, and half the team scored lower than the mean. It represents the "average" score. If each golfer had scored the same, they each would have scored the mean.

The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean average and the median. To calculate the mean weight of 50 people, add the 50 weights together and divide by To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts.

The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center. The substitution of one word for the other is common practice. When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values.

One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random. To see that both ways of calculating the mean are the same, consider the sample:. For example, if the total number of data values is 97, then. The median is the 49 th value in the ordered data. If the total number of data values is , then. The median occurs midway between the 50 th and 51 st values. The location of the median and the value of the median are not the same.

The next example illustrates the location of the median and the value of the median. AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows smallest to largest :.

Starting at the smallest value, the median is located between the 20 th and 21 st values the two 24s :. Press VarStats. The following data show the number of months patients typically wait on a transplant list before getting surgery. The data are ordered from smallest to largest.

Calculate the mean and median. Median: Starting at the smallest value, the median is the 20 th term, which is 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. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest.

A data set with two modes is called bimodal. The number of books checked out from the library from 25 students are as follows:. 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.

The mode can be calculated for qualitative data as well as for quantitative data. For example, if the data set is: red, red, red, green, green, yellow, purple, black, blue, the mode is red. 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. Five credit scores are , , , , Consider the annual earnings of workers at a factory.

This is discussed in more detail later in the text. 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. When only grouped data is available, you do not know the individual data values we only know intervals and interval frequencies ; therefore, you cannot compute an exact mean for the data set.

What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies.

To calculate the mean from a grouped frequency table we can apply the basic definition of mean:. We simply need to modify the definition to fit within the restrictions of a frequency table. Since we do not know the individual data values we can instead find the midpoint of each interval. The midpoint is. Find the best estimate of the class mean. Maris conducted a study on the effect that playing video games has on memory recall.

As part of her study, she compiled the following data:. What is the best estimate for the mean number of hours spent playing video games? Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers.

The mean and the median can be calculated to help you find the "center" of a data set. The mean is the best estimate for the actual data set, but the median is the best measurement when a data set contains several outliers or extreme values. The mode will tell you the most frequently occuring datum or data in your data set.

The mean, median, and mode are extremely helpful when you need to analyze your data, but if your data set consists of ranges which lack specific values, the mean may seem impossible to calculate. However, the mean can be approximated if you add the lower boundary with the upper boundary and divide by two to find the midpoint of each interval.

Multiply each midpoint by the number of values found in the corresponding range. Divide the sum of these values by the total number of data values in the set. The most frequent lengths are 25 and 27, which occur three times.

Use the following information to answer the next three exercises: 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.

Calculate the following:. Javier and Ercilia are supervisors at a shopping mall. Each was given the task of estimating the mean distance that shoppers live from the mall. They each randomly surveyed shoppers.

The samples yielded the following information. Use the following information to answer the next three exercises : We are interested in the number of years students in a particular elementary statistics class have lived in California. The information in the following table is from the entire section.

Calculator To find the mean and the median: Clear list L1. Enter 2nd 1 for list L1. Enter data into the list editor. Put the data values into list L1. Press the down and up arrow keys to scroll. Answer The most frequent score is 72, which occurs five times. Answer The most frequent number of books is 7, which occurs four times.

Sampling Distributions and Statistic of a Sampling Distribution You can think of a sampling distribution as a relative frequency distribution with a great many samples. Calculating the Mean of Grouped Frequency Tables When only grouped data is available, you do not know the individual data values we only know intervals and interval frequencies ; therefore, you cannot compute an exact mean for the data set.

Grade Interval Number of Students 50— Answer Find the midpoint of each interval, multiply by the corresponding number of teenagers, add the results and then divide by the total number of teenagers The midpoints are 1. Review The mean and the median can be calculated to help you find the "center" of a data set. Exercise 2. Grade Frequency Use the following information to answer the next three exercises: The following data show the lengths of boats moored in a marina. The data are ordered from smallest to largest: 16; 17; 19; 20; 20; 21; 23; 24; 25; 25; 25; 26; 26; 27; 27; 27; 28; 29; 30; 32; 33; 33; 34; 35; 37; 39; Answer The most frequent lengths are 25 and 27, which occur three times.

Bringing It Together Exercise 2.

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