Topic 10: Measure of central tendency

Measure of central tendency

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.

Formula to find MEAN, MEDIAN AND MODE

To find the mean, add up the values in the data set and then divide by the number of values that you added. 

To find the median, list the values of the data set in numerical order and identify which value appears in the middle of the list. 

To find the mode, identify which value in the data set occurs most often.

EXAMPLE TO FIND MEAN, MEDIAN AND MODE:  

1. Find the mean, median, mode, and range for the following list of values:

13, 18, 13, 14, 13, 16, 14, 21, 13

The mean is the usual average, so I'll add and then divide:

(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

Note that the mean, in this case, 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.

The median is the middle value, so first I'll have to rewrite the list in numerical 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.

The mode is the number that is repeated more often than any other, so 13 is the mode.

The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8.

mean: 15
median: 14
mode: 13
range: 8

Note: The formula for the place to find the median is "([the number of data points] + 1) ÷ 2", but you don't have to use this formula. You can just count in from both ends of the list until you meet in the middle, if you prefer, especially if your list is short. Either way will work.

2. Find the mean, median, mode, and range for the following list of values:

1, 2, 4, 7

The mean is the usual average:

(1 + 2 + 4 + 7) ÷ 4 = 14 ÷ 4 = 3.5

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. Because of this, the median of the list will be the mean (that is, the usual average) of the middle two values within the list. The middle two numbers are 2 and 4, so:

(2 + 4) ÷ 2 = 6 ÷ 2 = 3

So the median of this list is 3, a value that isn't in the list at all.

The mode is the number that is repeated most often, but all the numbers in this list appear only once, so there is no mode.

The largest value in the list is 7, the smallest is 1, and their difference is 6, so the range is 6.

mean: 3.5
median: 3
mode: none
range: 6

The values in the list above were all whole numbers, but the mean of the list was a decimal value. Getting a decimal value for the mean (or 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.

3. A student has gotten the following grades on his tests: 87, 95, 76, and 88. He wants an 85 or better overall. What is the minimum grade he must get on the last test in order to achieve that average?

The minimum grade is what I need to find. To find the average of all his grades (the known ones, plus the unknown one), I have to add up all the grades, and then divide by the number of grades. Since I don't have a score for the last test yet, I'll use a variable to stand for this unknown value: "x". Then computation to find the desired average is:

(87 + 95 + 76 + 88 + x) ÷ 5 = 85

Multiplying through by 5 and simplifying, I get:

87 + 95 + 76 + 88 + x = 425

346 + x = 425

x = 79

He needs to get at least a 79 on the last test.

SEE THIS VIDEO! (learn something from this video).....

Try to answer this Questions?

1) What is the mode of the following numbers?

$1, 2, 4, 6, 4$

2) What is the mode of the following numbers?

$8, 10, 8, 5, 4, 7, 5, 10, 8$

3) What is the mode of the following numbers?

$8, 1, 2, 6, 1, 4$

$4)\; What\; is\; the\; median\; of\; the\; following\; numbers?$

$5)\; What\; is\; the\; median\; of\; the\; following\; numbers?$

$6)\; What\; is\; the\; arithmetic\; mean\; of\; the\; following\; numbers?$

$7)\; What\; is\; the\; arithmetic\; mean\; of\; the\; following\; numbers$

$4,\; 10,\; 7,\; 4,\; 6$