Question

If \(\mathrm{L}=39 \cdot 5, \Delta_{1}=6, \Delta_{2}=9\) and \(\mathrm{c}=10\), then find the mode of the data. (1) \(45 \cdot 5\) (2) \(43 \cdot 5\) (3) \(46 \cdot 5\) (4) \(44 \cdot 5\)

Short answer

Answer: The mode of the data is 43 * 5.

Step-by-step solution

Identify given values

The given values are: \(L = 39 \cdot 5,\) \(\Delta_1 = 6,\) \(\Delta_2 = 9,\) \(c = 10\)

Apply the formula

Use the formula to find the mode: $$Mode = L + \frac{\Delta_1}{\Delta_1 + \Delta_2} \cdot c$$

Substitute the values and solve

Plug the given values into the formula: $$Mode = 39 \cdot 5 + \frac{6}{6 + 9} \cdot 10$$ Simplify the equation: $$Mode = 39 \cdot 5 + \frac{6}{15} \cdot 10$$ Now, calculate the value: $$Mode = 39 \cdot 5 + \frac{2}{5} \cdot 10$$ $$Mode = 39 \cdot 5 + 4$$

Evaluate the result

Add up the values to get the mode: $$Mode = (39 + 4) \cdot 5$$ $$Mode = 43 \cdot 5$$ The mode of the data is \(43 \cdot 5\), which corresponds to option (2).

Key concepts

mode calculation

In mathematics and statistics, finding the mode of a set is a crucial skill. The mode is the value that appears most frequently within a data set.
The mode is significant because it represents the most common occurrence, which can be particularly useful in understanding trends within datasets. To determine the mode in a grouped data using a class interval, you can use a specific formula. The formula is:

• Mode = \( L + \frac{\Delta_1}{\Delta_1 + \Delta_2} \cdot c \)

Here's what each symbol means:

• \( L \) is the lower limit of the modal class (the class with the highest frequency).
• \( \Delta_1 \) is the frequency difference between the modal class and the class before it.
• \( \Delta_2 \) is the frequency difference between the modal class and the class after it.
• \( c \) is the class interval width, the difference between the upper limit and lower limit of any class in the frequency distribution.

Substituting values into this formula correct calculation guided us to the result of \( 43 \cdot 5 \), which was the mode for the given problem.

class interval

Understanding class intervals is fundamental when dealing with group data. A class interval is a range of values in a frequency distribution into which data points are grouped. Each class interval has an upper and lower limit, ensuring that every value fits into at least one interval.
Here are the essentials you need to know about class intervals:

• The width of a class interval \(c\) is determined by subtracting the lower limit from the upper limit.
• Choosing equal class intervals makes calculations straightforward.
• Class intervals help aggregate data into manageable sections, making it easier to analyze.

In the given problem, the class interval width \(c\) was 10. This value is crucial for calculating the mode with the provided formula. By understanding the boundaries of each class, you can accurately compute statistical measures like the mode.

frequency distribution

Frequency distribution is an arrangement of statistical data that shows the number of observations within a specific interval. It's a helpful way to visualize data patterns, offering insights into how data values are spread over different intervals. A frequency distribution reveals important patterns such as:

• Where data values tend to cluster (seeing peaks and valleys can show popular ranges).
• Gaps in data that might indicate unique trends or missing data points.
• Overall data shape, which helps in determining modes, medians, and means.

In mathematical statistics, these insights aid significantly in problem-solving and decision-making. Moreover, frequency distributions simplify the process of calculating modes because they indicate which data range, or class interval, houses most data points.
In our problem, the frequency distribution was essential to identifying the modal class and accurately calculating the mode using the frequency values.