Find the mode from the given information:
| x | 13 | 14 | 15 | 16 | 17 | Total |
|---|---|---|---|---|---|---|
| f | x-1 | x-1 | 2x | x+1 | x+1 | 12 |
Answer (2)
To determine the mode from the frequency distribution, we first find the value of x that satisfies the total frequency of 12. After calculating the frequencies using x = 2 , we find that the highest frequency corresponds to x = 15 . Therefore, the mode is 15.
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Set up an equation by summing the frequencies and equating it to the total frequency: ( x − 1 ) + ( x − 1 ) + 2 x + ( x + 1 ) + ( x + 1 ) = 12 .
Solve the equation for x , which gives x = 2 .
Substitute the value of x into the expressions for the frequencies.
Determine the largest frequency and the corresponding value of x , which is the mode. The mode is 15 .
Explanation
Understanding the Problem We are given a frequency distribution table where the frequencies are expressed in terms of a variable x . Our goal is to find the mode, which is the value of the variable with the highest frequency. First, we need to find the value of x .
Setting up the Equation The sum of the frequencies must equal the total frequency, which is given as 12. So, we can write the equation: ( x − 1 ) + ( x − 1 ) + 2 x + ( x + 1 ) + ( x + 1 ) = 12
Solving for x Now, let's simplify and solve the equation for x :
x − 1 + x − 1 + 2 x + x + 1 + x + 1 = 12 6 x = 12 x = 6 12 x = 2
Calculating the Frequencies Now that we have the value of x , we can find the frequencies for each value in the table:
For x = 13 , the frequency is x − 1 = 2 − 1 = 1
For x = 14 , the frequency is x − 1 = 2 − 1 = 1
For x = 15 , the frequency is 2 x = 2 ( 2 ) = 4
For x = 16 , the frequency is x + 1 = 2 + 1 = 3
For x = 17 , the frequency is x + 1 = 2 + 1 = 3
Creating the Frequency Table Now we can create the frequency table with the calculated frequencies:
x
13
14
15
16
17
f
1
1
4
3
3
Determining the Mode The mode is the value of x with the highest frequency. From the table, we can see that the highest frequency is 4, which corresponds to x = 15 .
Final Answer Therefore, the mode is 15.
Examples
Understanding mode is useful in many real-life scenarios. For example, if you are analyzing the sales data of different products in a store, the mode would represent the product that is sold most frequently. This information can help the store owner make decisions about inventory management, marketing strategies, and product placement. Similarly, in education, if you are analyzing the test scores of students, the mode would represent the most common score, which can provide insights into the overall performance of the class.
