Find the mean, median, mode, and range of the following data set:
a). 2, 3, 4, 3, 5, 5, 6, 7, 8, 9, 6, 6, 5, 3
b). 13, 7, 8, 8, 2, 9, 11, 7, 8, 4, 5
c). 45, 48, 60, 42, 53, 47, 51, 54, 49, 48, 47, 53, 48, 44, 46.
Answer
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Hint: To find mean, median, mode, and range of the given data sets we will first define each of them and then by using definition and their formula will calculate mean, median, mode, and range.
Mean of ungrouped data is given by : $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
Median of a data set is given by ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term, if the total number of terms in the set is odd and also arranged in increasing order.
And, by $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$, if the total number of terms in the set is even and also arranged in increasing order.
Mode of a data set is the number that occurs most frequently in the given data set.
And, the range of the data set is equal to the difference between the maximum and the minimum value of the given data set.
Complete step-by-step answer:
Let us first understand the concept of mean, median, mode and range of the set of elements before calculating their values for the set given in the question.
Mean of a given data set is the average of all the terms present in the given data set.
So, Mean is given by: $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
Median of a given data set is the value which lies exactly in the middle of the given set when we arrange the given set in increasing order.
And, median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term, if the total number of terms in the set is odd.
And, by $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$, if the total number of terms in the set is even.
Mode of the given data set is the term which occurs the maximum number of times in the data set.
And, Range of the data set is equal to the difference between the maximum and the minimum value of the given data set.
Now, by using the above definition and formula we will find the mean, median, and mode of the given data sets.
a). From the question we can see that the data set is 2, 3, 4, 3, 5, 5, 6, 7, 8, 9, 6, 6, 5, 3.
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{2+3+4+3+5+5+6+7+8+9+6+6+5+3.}{14}\] = $\dfrac{67}{14}$ = 4.785
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9
We can see that total number of terms in the above data is 14, which is even and we also that when number of terms is even then median is given by: $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$
So, median = $\dfrac{{{\left( \dfrac{14}{2} \right)}^{th}}term+{{\left( \dfrac{14}{2}+1 \right)}^{th}}term}{2}$ = $\dfrac{{{\left( 7 \right)}^{th}}term+{{\left( 8 \right)}^{th}}term}{2}$ = $\dfrac{5+5}{2}$ = 5.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9 we can see that 3, 5, and 6 all three occur 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 3, 5, 6.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, we can see that 2 is the minimum term and 9 is the maximum term so range is equal to (9 - 2) = 7.
Hence, mean = 4.785, median = 5, mode = 3, 5, 6, and range = 7 is our required answer.
b). From the question we can see that the data set is 13, 7, 8, 8, 2, 9, 11, 7, 8, 4, 5
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{13+7+8+8+2+9+11+7+8+4+5}{11}\] = $\dfrac{67}{14}$ = 7.454
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13. We can see that total number of terms in the above data is 11, which is odd and we also that when number of terms is odd then median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$
So, median = ${{\left( \dfrac{11+1}{2} \right)}^{th}}$ = ${{\left( \dfrac{12}{2} \right)}^{th}}$ = ${{6}^{th}}term$ = 8.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13 we can see that 8 occurs 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 8.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13, we can see that 2 is the minimum term and 13 is the maximum term so range is equal to (13 - 2) = 11.
Hence, mean = 7.454, median = 8, mode = 8, and range = 11 is our required answer.
c). From the question we can see that the data set is 45, 48, 60, 42, 53, 47, 51, 54, 49, 48, 47, 53, 48, 44, 46.
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{45+48+60+42+53+47+51+54+49+48+47+53+48+44+46.}{15}\] = $\dfrac{735}{15}$ = 49
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60.
. We can see that total number of terms in the above data is 15, which is odd and we also that when number of terms is odd then median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$
So, median = ${{\left( \dfrac{15+1}{2} \right)}^{th}}$ = ${{\left( \dfrac{16}{2} \right)}^{th}}$ = ${{8}^{th}}term$ = 48.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60, we can see that 48 occurs 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 48.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60, we can see that 42 is the minimum term and 60 is the maximum term so range is equal to (60 - 42) = 18.
Hence, mean = 49, median = 48, mode = 48, and range = 18 is our required answer.
Note: While calculating the median of the given data set, it is very much necessary to arrange the given data set in increasing order, otherwise students will not get the correct answer and there is so much chance of making mistakes which writing sets frequently, so students are required to take care of that.
Mean of ungrouped data is given by : $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
Median of a data set is given by ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term, if the total number of terms in the set is odd and also arranged in increasing order.
And, by $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$, if the total number of terms in the set is even and also arranged in increasing order.
Mode of a data set is the number that occurs most frequently in the given data set.
And, the range of the data set is equal to the difference between the maximum and the minimum value of the given data set.
Complete step-by-step answer:
Let us first understand the concept of mean, median, mode and range of the set of elements before calculating their values for the set given in the question.
Mean of a given data set is the average of all the terms present in the given data set.
So, Mean is given by: $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
Median of a given data set is the value which lies exactly in the middle of the given set when we arrange the given set in increasing order.
And, median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$term, if the total number of terms in the set is odd.
And, by $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$, if the total number of terms in the set is even.
Mode of the given data set is the term which occurs the maximum number of times in the data set.
And, Range of the data set is equal to the difference between the maximum and the minimum value of the given data set.
Now, by using the above definition and formula we will find the mean, median, and mode of the given data sets.
a). From the question we can see that the data set is 2, 3, 4, 3, 5, 5, 6, 7, 8, 9, 6, 6, 5, 3.
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{2+3+4+3+5+5+6+7+8+9+6+6+5+3.}{14}\] = $\dfrac{67}{14}$ = 4.785
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9
We can see that total number of terms in the above data is 14, which is even and we also that when number of terms is even then median is given by: $\dfrac{{{\left( \dfrac{n}{2} \right)}^{th}}term+{{\left( \dfrac{n}{2}+1 \right)}^{th}}term}{2}$
So, median = $\dfrac{{{\left( \dfrac{14}{2} \right)}^{th}}term+{{\left( \dfrac{14}{2}+1 \right)}^{th}}term}{2}$ = $\dfrac{{{\left( 7 \right)}^{th}}term+{{\left( 8 \right)}^{th}}term}{2}$ = $\dfrac{5+5}{2}$ = 5.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9 we can see that 3, 5, and 6 all three occur 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 3, 5, 6.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 2, 3, 3, 3, 4, 5, 5, 5, 6, 6, 6, 7, 8, 9, we can see that 2 is the minimum term and 9 is the maximum term so range is equal to (9 - 2) = 7.
Hence, mean = 4.785, median = 5, mode = 3, 5, 6, and range = 7 is our required answer.
b). From the question we can see that the data set is 13, 7, 8, 8, 2, 9, 11, 7, 8, 4, 5
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{13+7+8+8+2+9+11+7+8+4+5}{11}\] = $\dfrac{67}{14}$ = 7.454
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13. We can see that total number of terms in the above data is 11, which is odd and we also that when number of terms is odd then median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$
So, median = ${{\left( \dfrac{11+1}{2} \right)}^{th}}$ = ${{\left( \dfrac{12}{2} \right)}^{th}}$ = ${{6}^{th}}term$ = 8.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13 we can see that 8 occurs 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 8.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 2, 4, 5, 7, 7, 8, 8, 8, 9, 11, 13, we can see that 2 is the minimum term and 13 is the maximum term so range is equal to (13 - 2) = 11.
Hence, mean = 7.454, median = 8, mode = 8, and range = 11 is our required answer.
c). From the question we can see that the data set is 45, 48, 60, 42, 53, 47, 51, 54, 49, 48, 47, 53, 48, 44, 46.
We will first find the mean of the given data. From the above definition we know that:
Mean = $\dfrac{\text{Sum of all the term in given set}}{\text{Total number of terms in that set}}$
So, we can say that mean = \[\dfrac{45+48+60+42+53+47+51+54+49+48+47+53+48+44+46.}{15}\] = $\dfrac{735}{15}$ = 49
Now, before calculating the median, mode and range we will first arrange the given data in ascending order.
So, after arranging the data set in ascending order we will get: 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60.
. We can see that total number of terms in the above data is 15, which is odd and we also that when number of terms is odd then median is given by: ${{\left( \dfrac{n+1}{2} \right)}^{th}}$
So, median = ${{\left( \dfrac{15+1}{2} \right)}^{th}}$ = ${{\left( \dfrac{16}{2} \right)}^{th}}$ = ${{8}^{th}}term$ = 48.
Now, we will calculate the mode. Since, from above definition of mode we know that mode is the number that occurs most frequently in the given data set.
So, in the data set 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60, we can see that 48 occurs 3 times in the given data and it is the maximum occurrence in the data set. Hence, mode is equal to 48.
Now, we will calculate the range of the given data. From the above definition we know that range is difference of greatest and lowest term in the set and in the set 42, 44, 45, 46, 47, 47, 48, 48, 48, 49, 51, 53, 53, 54, 60, we can see that 42 is the minimum term and 60 is the maximum term so range is equal to (60 - 42) = 18.
Hence, mean = 49, median = 48, mode = 48, and range = 18 is our required answer.
Note: While calculating the median of the given data set, it is very much necessary to arrange the given data set in increasing order, otherwise students will not get the correct answer and there is so much chance of making mistakes which writing sets frequently, so students are required to take care of that.
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