Question

For the dataset {2, 3, 3, 5, 7, 10}, find the mean, median, and mode.

Answer 1

Of course! Let's go through each calculation step-by-step, explaining the concept behind each measure of central tendency.

The dataset we are working with is: {2, 3, 3, 5, 7, 10}

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1. The Mean (The Average)

What is the Mean?

The mean is the most common type of average. It represents the "balancing point" of the data. If you were to take all the values in the dataset and distribute them equally among all the data points, the mean is the value that each point would have.

How to Calculate the Mean:

There are two simple steps:
1. Sum: Add up all the numbers in the dataset.
2. Divide: Divide that sum by the total count of numbers in the dataset.

Applying it to our dataset: {2, 3, 3, 5, 7, 10}

• Step 1: Sum the numbers.
  2 + 3 + 3 + 5 + 7 + 10 = 30
  The sum of all the values is 30.

• Step 2: Count the numbers.
  There are 6 individual numbers in the dataset.

• Step 3: Divide the sum by the count.
  Mean = 30 / 6 = 5

Therefore, the mean of the dataset is 5.

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2. The Median (The Middle Value)

What is the Median?

The median is the value that is physically in the middle of the dataset after it has been arranged in order from least to greatest. It is a useful measure because it is not affected by unusually high or low values (called outliers).

How to Calculate the Median:

1. Order: Arrange the dataset in numerical order (least to greatest).
2. Find the Middle:
   • If there is an odd number of data points, the median is the single number in the exact middle.
   • If there is an even number of data points (like in our case), there will be two middle numbers. The median is the average of these two numbers.

Applying it to our dataset: {2, 3, 3, 5, 7, 10}

• Step 1: Order the dataset.
  Our dataset is already in order: {2, 3, 3, 5, 7, 10}

• Step 2: Find the middle.
  There are 6 numbers (an even amount). This means we don't have one single middle number. Instead, we have a pair of middle numbers. To find them, we can cross off numbers from each end until we are left with the middle pair:

  {2, 3, **3**, **5**, 7, 10}

  The two middle numbers are 3 and 5.

• Step 3: Calculate the average of the two middle numbers.
  We add the two middle numbers together and divide by 2.
  (3 + 5) / 2 = 8 / 2 = 4

Therefore, the median of the dataset is 4.

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3. The Mode (The Most Frequent)

What is the Mode?

The mode is the simplest of the three measures. It is the number that appears most frequently in the dataset. A dataset can have one mode, more than one mode (if multiple numbers are tied for the highest frequency), or no mode at all (if every number appears only once).

How to Find the Mode:

1. Scan: Look through the dataset.
2. Count: Tally how many times each unique number appears.
3. Identify: The number that appears most often is the mode.

Applying it to our dataset: {2, 3, 3, 5, 7, 10}

• Step 1: Let's count the frequency of each number.
  2 appears 1 time
  3 appears 2 times
  5 appears 1 time
  7 appears 1 time
  * 10 appears 1 time

• Step 2: Identify the number with the highest frequency.
  The number 3 appears more often than any other number in the set.

Therefore, the mode of the dataset is 3.

Final Summary

For the dataset {2, 3, 3, 5, 7, 10}:
Mean: 5
Median: 4
* Mode: 3