MEASURES OF CENTRAL LOCATION
MEASURES OF CENTRAL LOCATION
1.1 Definition of Mean, Median & Model Whether you are a math student, survey taker, statistician or researcher, you are going to need to calculate the average of multiple numbers from time to time. But finding the average is not always straightforward. Averages can be found in three ways--mean, median and mode. Mean When you think of averaging, you are most likely to think of finding the mean. You add all of the numbers in the set and divide by how many numbers are in the list. is the sum of all terms divided by the total number of terms. Find the mean, median for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The mean is the usual average, so: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15 Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers. Median (Md) is the middle value if the set of observation is odd, and the mean of the middle two values when data is even that is arranged from highest to lowest or vice versa. Find the median for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The median is the middle value, so I'll have to rewrite the list in order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number: 13, 13, 13, 13, 14, 14, 16, 18, 21 So the median is 14. Find the median for the following list of values: 1, 2, 5, 7 The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers. In this case, the median is the mean (the usual average) of the middle two values: (2 + 5) ÷ 2 = 7 ÷ 2 = 3. •The list values were whole numbers, but the median was a decimal value. Getting a decimal value for the median, if you have an even number of data points is perfectly okay; don't round your answers to try to match the format of the other numbers. Mode the value that appears the most. The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list. The mode is the number that is repeated more often than any other. Find the mode for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The mode is the number that is repeated more often than any other, so 13 is the mode. 1.2 Central Tendency The term central tendency refers to the "middle" value or perhaps a typical value of the data, and is measured using the mean, median, or mode. Each of these measures is calculated differently, and the one that is best to use depends upon the situation. Mean The mean is the most commonly-used measure of central tendency. When we talk about an "average", we usually are referring to the mean. The mean is simply the sum of the values divided by the total number of items in the set. The result is referred to as the arithmetic mean. Sometimes it is useful to give more weighting to certain data points, in which case the result is called the weighted arithmetic mean. The notation used to express the mean depends on whether we are talking about the population mean or the sample mean The mean is valid only for interval data or ratio data. Since it uses the values of all of the data points in the population or sample, the mean is influenced by outliers that may be at the extremes of the data set. Median The median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. There is an equal number of points above and below the median. For example, in the data set {1,2,3,4,5} the median is 3; there are two data points greater than this value and two data points less than this value. In this case, the median is equal to the mean. But consider the data set {1,2,3,4,10}. In this dataset, the median still is three, but the mean is equal to 4. If there is an even number of data points in the set, then there is no single point at the middle and the median is calculated by taking the mean of the two middle points. The median can be determined for ordinal data. Unlike the mean, the median is not influenced by outliers at the extremes of the data set. For this reason, the median often is used when there are a few extreme values that could greatly influence the mean and distort what might be considered typical. This often is the case with home prices and with income data for a group of people, which often is very skewed. For such data, the median often is reported instead of the mean. For example, in a group of people, if the salary of one person is 10 times the mean, the mean salary of the group will be higher because of the unusually large salary. In this case, the median may better represent the typical salary level of the group. Mode The mode is the most frequently occurring value in the data set. For example, in the data set {1,2,3,4,4}, the mode is equal to 4. A data set can have more than a single mode, in which case it is multimodal. In the data set {1,1,2,3,3} there are two modes: 1 and 3. The mode can be very useful for dealing with categorical data. For example, if a sandwich shop sells 10 different types of sandwiches, the mode would represent the most popular sandwich. The mode also can be used with ordinal, interval, and ratio data. However, in interval and ratio scales, the data may be spread thinly with no data points having the same value. In such cases, the mode may not exist or may not be very meaningful. When to use Mean, Median, and Mode
Things to Remember •There are three kinds of averages used in representing a typical value in a set of observations. •The kind of average to use in a set of data is determined by the nature of characteristics of the set. •The mean is associated with the interval/ ratio data, the median with ordinal data, and the mode with nominal data. •It is possible for a distribution to have more than one mode . But a distribution has only one mean and one median. •The mean and the median need not to be a value in a distribution. However, the mode should be one of the numbers in the distribution. Sample Application of Mean in a Research Study FREQUENCY DISTRIBUTION OF ATTITUDES OF STUDENTS TOWARDS MATHEMATICS
1- Strongly Disagree 2- Disagree 3- Undecided 4-Agree 5-Strongly Disagree
Steps in solving Mean, Median and Mode using MS Excel For example we have the following raw data representing the age at first marriage of a sample of 12 adults. 18 18 19 19 19 19 20 20 20 21 21 22 22 23 23 24 25 26 26 27 27 29 30 31 1. Enter the scores in one of the columns or rows on the Excel spreadsheet. After the data has been entered, place the cursor where you wish to have the mean (average), median or mode appear and click the mouse button.  2. You can either type in the dialogue box what measure of central location you will use and enter the cell range for your list of numbers. For example solving for the mean of the given data, we enter in the dialogue box “=averageB2:Y2”. |
1.1 Definition of Mean, Median & Model
Whether you are a math student, survey taker, statistician or researcher, you are going to need to calculate the average of multiple numbers from time to time. But finding the average is not always straightforward. Averages can be found in three ways--mean, median and mode.
Mean
When you think of averaging, you are most likely to think of finding the mean. You add all of the numbers in the set and divide by how many numbers are in the list.
is the sum of all terms divided by the total number of terms.
Find the mean, median for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
13, 18, 13, 14, 13, 16, 14, 21, 13
The mean is the usual average, so:
(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15
Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers.
Median (Md)
is the middle value if the set of observation is odd, and the mean of the middle two values when data is even that is arranged from highest to lowest or vice versa.
Find the median for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
13, 18, 13, 14, 13, 16, 14, 21, 13
The median is the middle value, so I'll have to rewrite the list in order:
13, 13, 13, 13, 14, 14, 16, 18, 21
There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number:
13, 13, 13, 13, 14, 14, 16, 18, 21
So the median is 14.
Find the median for the following list of values:
1, 2, 5, 7
1, 2, 5, 7
The median is the middle number. In this example, the numbers are already listed in numerical order, so I don't have to rewrite the list. But there is no "middle" number, because there are an even number of numbers. In this case, the median is the mean (the usual average) of the middle two values: (2 + 5) ÷ 2 = 7 ÷ 2 = 3.
•The list values were whole numbers, but the median was a decimal value. Getting a decimal value for the median, if you have an even number of data points is perfectly okay; don't round your answers to try to match the format of the other numbers.
Mode
the value that appears the most.
The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.
The mode is the number that is repeated more often than any other.
Find the mode for the following list of values:
13, 18, 13, 14, 13, 16, 14, 21, 13
13, 18, 13, 14, 13, 16, 14, 21, 13
The mode is the number that is repeated more often than any other, so 13 is the mode.
1.2 Central Tendency
The term central tendency refers to the "middle" value or perhaps a typical value of the data, and is measured using the mean, median, or mode. Each of these measures is calculated differently, and the one that is best to use depends upon the situation.
Mean
The mean is the most commonly-used measure of central tendency. When we talk about an "average", we usually are referring to the mean. The mean is simply the sum of the values divided by the total number of items in the set. The result is referred to as the arithmetic mean. Sometimes it is useful to give more weighting to certain data points, in which case the result is called the weighted arithmetic mean.
The notation used to express the mean depends on whether we are talking about the population mean or the sample mean
The mean is valid only for interval data or ratio data. Since it uses the values of all of the data points in the population or sample, the mean is influenced by outliers that may be at the extremes of the data set.
Median
The median is determined by sorting the data set from lowest to highest values and taking the data point in the middle of the sequence. There is an equal number of points above and below the median. For example, in the data set {1,2,3,4,5} the median is 3; there are two data points greater than this value and two data points less than this value. In this case, the median is equal to the mean. But consider the data set {1,2,3,4,10}. In this dataset, the median still is three, but the mean is equal to 4. If there is an even number of data points in the set, then there is no single point at the middle and the median is calculated by taking the mean of the two middle points.
The median can be determined for ordinal data. Unlike the mean, the median is not influenced by outliers at the extremes of the data set. For this reason, the median often is used when there are a few extreme values that could greatly influence the mean and distort what might be considered typical. This often is the case with home prices and with income data for a group of people, which often is very skewed. For such data, the median often is reported instead of the mean. For example, in a group of people, if the salary of one person is 10 times the mean, the mean salary of the group will be higher because of the unusually large salary. In this case, the median may better represent the typical salary level of the group.
Mode
The mode is the most frequently occurring value in the data set. For example, in the data set {1,2,3,4,4}, the mode is equal to 4. A data set can have more than a single mode, in which case it is multimodal. In the data set {1,1,2,3,3} there are two modes: 1 and 3.
The mode can be very useful for dealing with categorical data. For example, if a sandwich shop sells 10 different types of sandwiches, the mode would represent the most popular sandwich. The mode also can be used with ordinal, interval, and ratio data. However, in interval and ratio scales, the data may be spread thinly with no data points having the same value. In such cases, the mode may not exist or may not be very meaningful.
When to use Mean, Median, and Mode
Measurement Scale | Best Measure of the "Middle" |
Nominal (Categorical) | Mode |
Ordinal | Median |
Interval | Mean |
Ratio | Mean |
Things to Remember
•There are three kinds of averages used in representing a typical value in a set of observations.
•The kind of average to use in a set of data is determined by the nature of characteristics of the set.
•The mean is associated with the interval/ ratio data, the median with ordinal data, and the mode with nominal data.
•It is possible for a distribution to have more than one mode . But a distribution has only one mean and one median.
•The mean and the median need not to be a value in a distribution. However, the mode should be one of the numbers in the distribution.
Sample Application of Mean in a Research Study
FREQUENCY DISTRIBUTION OF ATTITUDES OF STUDENTS TOWARDS MATHEMATICS
Attitudes towards Mathematics | 1 | 2 | 3 | 4 | 5 | WM | Description |
1. Mathematics is my favorite subject | 1 | 2 | 3 | 4 | 5 | WM | Description |
2. I would like to work in Mathematics related field. | 6 | 3 | 8 | 25 | 8 | 3.52 | Agree |
3. I think Mathematics is challenging. | 4 | 7 | 9 | 16 | 14 | 3.58 | Agree |
4. I am always anxious in my mathematics class. | 6 | 9 | 10 | 10 | 15 | 3.38 | Undecided |
6. I find mathematics boring and doll. | 12 | 13 | 20 | 3 | 2 | 2.4 | Disagree |
7. I solve other exercises in mathematics aside from the assigned ones. | 14 | 16 | 10 | 5 | 5 | 2.42 | Disagree |
8. Mathematics trains me to be systematic. | 10 | 7 | 13 | 15 | 5 | 2.96 | Undecided |
9. My mind travels far when I am in my mathematics class. | 2 | 1 | 25 | 12 | 10 | 3.54 | Agree |
10. Studying mathematics deprives me of the chance to attend to more fruitful and enjoyable activities. | 10 | 15 | 8 | 10 | 7 | 2.78 | Undecided |
1- Strongly Disagree 2- Disagree 3- Undecided 4-Agree
5-Strongly Disagree
Five Point Scale | Description |
4.5 – 5 | Strongly Agree |
3.5 – 4.49 | Agree |
2.5 – 3.49 | Undecided |
1.5 – 2.49 | Disagree |
1-1.49 | Strongly Disagree |
Steps in solving Mean, Median and Mode using MS Excel
For example we have the following raw data representing the age at first marriage of a sample of 12 adults.
18 18 19 19 19 19 20 20 20 21
21 22 22 23 23 24 25 26 26 27
27 29 30 31
1. Enter the scores in one of the columns or rows on the Excel spreadsheet. After the data has been entered, place the cursor where you wish to have the mean (average), median or mode appear and click the mouse button.
2. You can either type in the dialogue box what measure of central location you will use and enter the cell range for your list of numbers. For example solving for the mean of the given data, we enter in the dialogue box “=averageB2:Y2”.
No comments:
Post a Comment