Lesson Video: The Median of a Data Set Mathematics • 6th Grade

In this video, we will learn how to find and interpret the median of a data set.

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Video Transcript

In this video, we will learn how to find and interpret the median of a data set. The median is a type of average. And we will begin by defining it. The median is an example of a measure of center or a measure of central tendency. The mode and mean are also measures of center, although in this video we will only discuss the median. The median is a single number which gives us some information about the typical values in a data set. Specifically, the median of a set of data represents the middle value. Half of the data is above the median and half of the data is below the median. We will begin by looking at some examples where we calculate the median of a small data set.

Find the median of the values six, eight, 16, six, and 19.

The median is the middle value of a set of data. In order to calculate the median of a small set of data, we follow two simple steps. Firstly, we put the numbers in ascending order. In this question, the order will be six, six, eight, 16, and 19. Our second step is to find the middle number or value. With a small data set, we can do this by crossing off numbers from either end. We begin by crossing off six and 19, the smallest and largest numbers. We then cross off six and 16. We are left with one number in the middle. Therefore, the median of the set of values is eight.

An alternative way of finding the median after putting the numbers in ascending order is to use a little rule or formula. The median position can be calculated using the formula ๐‘› plus one divided by two, where ๐‘› is the number of values. In this question, we have five numbers. Therefore, the median position can be calculated by adding five to one and then dividing by two. This is equal to three. The median number will therefore be the third number in the list. Counting the numbers from left to right in ascending order, we see that the third number is eight. Therefore, the median is eight.

Find the median of the values 13, five, nine, 10, two, and 15.

We can calculate the median of any data set by following two steps. Firstly, we put the numbers in ascending order. In this question, our six numbers in ascending order are two, five, nine, 10, 13, and 15. The median is the middle number. Therefore, we need to find the middle value from our list. One way to do this is to cross off a number from each end of the list. We cross off the highest number, 15, and the lowest number, two. We then cross off the next highest and next lowest, 13 and five. We are now left with two middle values, nine and 10.

To find the median in this case, we find the number that is halfway between the middle values. This can be calculated by adding the two middle values and then dividing by two. Nine plus 10 is equal to 19, and dividing this by two gives us 9.5. The median of the set of six values is therefore equal to 9.5. Half of our values must be above this, in this case, 10, 13, and 15. And half of the values must be below 9.5, nine, five, and two.

An alternative method here wouldโ€™ve been to have found the median position. We do this using the formula ๐‘› plus one divided by two, where ๐‘› is the number of values. In this question, we had six values. We need to add six to one and divide by two. This is equal to 3.5. The median will therefore lie between the third and fourth value. As the third value was equal to nine and the fourth value 10, once again, we have proved that the median was 9.5.

What is the median of the following numbers: 11, 11, eight, eight, nine, and nine?

The median of any data set is the middle number. We can calculate this using two steps. Firstly, we put the numbers in ascending order. In this question, we have two eights, followed by two nines, followed by two 11s. For a small data set like this, we can then find the middle number by crossing off one number from either end, firstly, the eight and the 11. We can then cross off the second eight and the second 11. We are left with two middle numbers. Normally, when there are two middle values, we have to calculate the number that is halfway between them. In this case, since both the middle numbers are the same, this value is the median. The median of the six numbers 11, 11, eight, eight, nine, and nine is nine.

The next question weโ€™ll look at is more complicated as there is a missing number, but we are given the median.

Jennifer has the following data: 10, eight, seven, nine, and ๐‘š. If the median is eight, which number could ๐‘š be? Is it (A) seven, (B) 8.5, (C) nine, (D) 9.5, or (E) 10?

The median represents the middle of the data. This means that half of the data values are above it and half are below it. We can start by putting the data in order and thinking about how we can work out the number ๐‘š. Writing the four values that we know in order gives us seven, eight, nine, and 10. We are told that the median is eight. As the median is the middle of five values, it must be the third number when listed in order from lowest to highest. There must be two values above eight and two values below eight. For the number to be written to the left of eight, it must be less than or equal to eight.

If ๐‘š was any integer value less than seven, it would be the first number in the list. Positive integers here could be one, two, three, four, five, and six. None of these are options in the question, though. The missing number could also be seven as both the sevens would be written to the left-hand side of eight as they are less than eight. Out of our five options, the correct answer is option (A), seven. ๐‘š could not be 8.5, nine, 9.5, or 10 as all of these are greater than eight and would be on the right-hand side of eight.

It is possible that the missing number couldโ€™ve been eight. If this was the case, seven would be the first number in our list. We would then have two eights. The third number would still be eight, which is the median. Whilst the value of ๐‘š could be any number less than or equal to eight, the correct answer in this case is seven.

The final question that weโ€™ll look at is more complicated as it involves a large data set.

Find the median of the set of data represented in this line plot.

The number line here goes from two to 14. And the number of crosses represents how many of each value we have. There are four crosses above the number four. Therefore, we have four fours. There are six crosses above the number five, so we have six fives. We have two sixes and two sevens. Continuing this, we have six eights, three nines, two 10s, three 11s, three 12s, six 13s, and three 14s. We know that the median is the middle value when the numbers are in ascending order.

One way to answer this question would be to write all the numbers out from smallest to largest. We would write the number four four times. We would write the number five six times. There would be two sixes and two sevens. The list would continue as shown all the way up to three 14s. There are a total here of 40 values. We could find the middle number by crossing off one from each end, firstly, the highest number and the lowest number. We repeat this by crossing another four and another 14. Crossing off the next 10 smallest numbers and next 10 largest numbers would leave us with the numbers from seven to 11. We could continue this process until weโ€™re left with two middle values, eight and nine.

When there are an even number of values in total, there will always be two middle values. We can then find the median by finding the number that is halfway between them. We do this in this case by adding eight and nine and then dividing the answer by two. Eight plus nine is equal to 17, and half of this is 8.5. Clearly, 8.5 is halfway between eight and nine. Therefore, this is the median of the set of data.

An alternative method here would be to calculate the median position first. The median position can be calculated using the formula ๐‘› plus one divided by two, where ๐‘› is the total number of values. 40 plus one is equal to 41. And dividing this by two gives us 20.5. As 20.5 lies between the integers 20 and 21, we know that the median will be halfway between the 20th and 21st value.

To find the 20th and 21st values, we can work out the running total or cumulative frequency. We do this by adding the number of values we have. Four plus six is equal to 10. Adding another two gives us 12. This means that 12 values are six or lower. Adding another two gives us 14, and adding six gives us 20. This means that there are 20 values that are eight or less. As there are 40 values in total, there must therefore be 20 values that are nine or greater. The 20th value is equal to eight, and the 21st value is equal to nine. Once again, finding the midpoint of these two values gives us a median of 8.5. This method is useful when we have a large number of values as it saves writing out the whole data set.

We will now summarize the key points from this video on the median of a set of data. In order to calculate the median from a data set, our first step is to write the data in ascending order from least to greatest. Next, we need to find the middle value or values. If there are an odd number of data values, there will be exactly one value in the middle. However, if there are an even number of data values, there will be two values in the middle. Our third and final step is to find the median.

If there are an odd number of data values, the median is the middle number. If there are an even number of data values, the median is halfway between the two middle numbers. Finally, if we have a large data set, we can calculate the median position using the formula ๐‘› plus one divided by two, where ๐‘› is the total number of data values.

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