Mean Median Mode And Standard Deviation

To find the mode, we look for the class with the highest frequency. Here, the age group 2-4 has the highest frequency, with 7 children. Thus, the mode class is 2-4. To calculate the mode, we use the following formula: where:
is the lower boundary of the modal class (2),
is the frequency of the modal class (7),
is the frequency of the class before the modal class (5),
is the frequency of the class after the modal class (3),
is the class width (2).
Substitute the values:

The correct option is (C):

First, arrange the data in ascending order:
The Mode is the most frequent value in the dataset. Here, the mode is 9, as it appears twice.
The Median is the middle value when the data is arranged in order. Since there are 10 data points (even number), the median is the average of the 5th and 6th values:
The Mean is the sum of all values divided by the number of values: Thus, the Mean = Median = Mode = 6.5.

The correct option is (C):

In this case, the organization is interested in finding the most popular TV serial, which refers to identifying the value that appears most frequently in the data. This corresponds to the , as the mode is the value that occurs most frequently in a dataset. The mean and median are not suitable measures for identifying the most popular option in this context. Hence, the preferred measure is .

The correct option is (B): Mode

Step 1: Recall the formula for the mean of the first natural numbers.

The sum of the first natural numbers is:

The mean of the first natural numbers is:

Step 2: Set the mean equal to 15 and solve for .

Multiply through by 2:

Solve for :

Final Answer: The value of is , which corresponds to option .

Step 1: Recall the formula for the median.

The median is the middle value of a data set when the number of observations is odd, or the average of the two middle values when the number of observations is even. Here, there are 10 observations (even), so the median is the average of the 5th and 6th values.

Step 2: Identify the 5th and 6th values in the data set.

From the given data set, the 5th value is , and the 6th value is .

The median is given as 34. Therefore:

Substitute the known values:

Step 3: Solve for .

Multiply through by 2 to eliminate the fraction:

Simplify:

Solve for :

Final Answer: The value of is , which corresponds to option .

Step 1: Understand the scenario.

The statement "most people are showing interest" implies that the red color is the most frequently chosen preference among car buyers. This indicates that the data has a clear peak or most frequent value.

Step 2: Recall the definitions of measures of central tendency.

• Mean: The average of all data points.
• Median: The middle value when data is arranged in ascending order.
• Mode: The value that appears most frequently in the data set.

In this case, the company is focusing on the "most frequent" choice, which aligns with the definition of the mode.

Final Answer: The measure of central tendency they selected is , which corresponds to option .

Step 1: Compute the total sum of the original observations.

The mean of the 100 observations is given as 49. The total sum of the observations can be calculated as:

Step 2: Identify the incorrect and correct values.

The incorrect values registered were , and the correct values should have been . The difference between the correct and incorrect values is:

Step 3: Adjust the total sum to account for the correction.

The corrected total sum is:

Step 4: Compute the corrected mean.

The corrected mean is:

Final Answer: The corrected mean is , which corresponds to option .

Step 1: Recall the empirical relationship between mean, median, and mode.

In moderately skewed distributions, there is an approximate empirical relationship:

Step 2: Express the distances mathematically.

Let the mean be , the median be , and the mode be . The distance between the mode and mean is:

The distance between the median and mean is:

Step 3: Use the empirical formula to relate these distances.

From the empirical relationship , we can write:

Taking absolute values:

Step 4: Compare the distances.

The problem states that the distance between the mode and mean is times the distance between the median and mean. From the above equation:

Comparing this with , we find:

Final Answer: The value of is , which corresponds to option .

Correct answer: Range

Explanation:
Measures of central tendency are statistical measures that describe the center of a data set. These include: - Mean: The average of the data set. - Median: The middle value when the data set is ordered. - Mode: The value that appears most frequently in the data set. The range, however, is a measure of the spread or dispersion of the data, not the central tendency. It is the difference between the largest and smallest values in the data set. Hence, the correct answer is Range, as it is not a measure of central tendency.

Correct answer: None of these

Explanation:
The mode of a data set is the value that appears most frequently. In the given data set: all the numbers appear only once, meaning there is no repetition of any value. Therefore, this data set does not have a mode. Hence, the mode of the data set is None of these.

In the given data, the mode is the number that appears most frequently. The data consists of the following values: The number 7 appears most frequently, and we are told that the mode is 7. To determine the value of , note that the term must not interfere with the frequency of 7. Thus, cannot be 7, otherwise it would alter the frequency. Now, to determine , we observe that the data must have at least 3 occurrences of 7 to make it the mode. Since there are already 3 sevens (7, 7, 7), we conclude that:

The correct option is (B):

Graphically, the median and mode can be determined using a cumulative frequency curve (ogive) or histogram. The median is found by locating the point corresponding to the 50th percentile in the cumulative frequency curve, and the mode is identified as the highest point of the histogram. However, the mean is an algebraic concept that requires the sum of the data values and the number of values, which cannot be directly obtained from a graphical representation. Thus, the mean cannot be determined graphically.

The correct option is (A): Mean

In a data set with an odd number of observations, the median is the middle observation when the data is arranged in ascending order.
Given that there are 25 observations, the median will be the -th observation, which is the 13th observation.

The correct option is (B): 13^th

The given table provides the following data: To find the mean, we use the formula for the mean of grouped data: where represents the frequency and represents the corresponding marks. Let's first calculate the sum of for each group:
For , the contribution is .
For , the contribution is .
For , the contribution is .
For , the contribution is .
For , the contribution is .
Thus, Now, let's calculate the sum of frequencies: The mean is given as , so: Simplifying: Now, multiply both sides by 15: Solving for :

The correct option is (C):

Let the sum of the 12 numbers be . The mean is given by: We are given that the mean is 19, so: Now, if 4 is subtracted from each number, the new sum of the numbers will be: The new mean is:

The correct option is (D):

In a frequency distribution, the relationship between mode, median, and mean can be given by the empirical formula: Given: - Mode = 5 - Median = 10 Substituting the values into the formula:

The correct option is (C):

We can use the empirical relationship between the mean, median, and mode: $ M $

Therefore, the median of the data is 48.