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Chapter 2: Problem 3

Find the mean, the median, and the mode for the sample $$ 11234 $$

Short Answer

Expert verified
Mean: 2.2, Median: 2, Mode: 1.

Step by step solution

01

List the Numbers

Start by listing the numbers given in the sample. In this case, the numbers are: \(1, 1, 2, 3, 4\).
02

Calculate the Mean

The mean is calculated by dividing the sum of all numbers by the number of numbers. First, add all the numbers: \(1 + 1 + 2 + 3 + 4 = 11\). Next, divide by the number of numbers, which is 5: \(\frac{11}{5} = 2.2\).
03

Find the Median

The median is the middle number when the numbers are in order. Since there are 5 numbers, the median is the 3rd number: \(2\).
04

Identify the Mode

The mode is the number that appears most frequently. In the list \(1, 1, 2, 3, 4\), the number \(1\) appears twice, making it the mode.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
Calculating the mean is a foundational step in descriptive statistics, often used to summarize a set of data with a single value representing the average. To find the mean, follow these simple steps:
  • First, add up all the numbers in your data set. This gives you the total sum of the numbers.

  • Next, count how many numbers are in the data set to get the total number of data points.

  • Finally, divide the total sum by the number of data points. This result is the mean.
In our example with the numbers 1, 1, 2, 3, and 4:
  • The sum is calculated as: \(1 + 1 + 2 + 3 + 4 = 11\).

  • There are 5 numbers in total.

  • Therefore, the mean is \(\frac{11}{5} = 2.2\).
The mean provides us with an idea about the central tendency of the data, balancing all data points.
Median Determination
Finding the median involves identifying the middle value in a sorted list of numbers. The median offers a useful measure of central tendency, particularly when the dataset includes outliers or skewed values.
  • First, arrange the numbers in ascending order, if they are not already.

  • If the number of data points is odd, the median is the center number.
  • If the number of data points is even, calculate the median as the average of the two middle numbers (this involves adding them together and diving by two).
For instance, with the numbers 1, 1, 2, 3, and 4, the count is odd:
  • The data set in order is \([1, 1, 2, 3, 4]\).

  • The third number in this sequence, 2, is the median.
The median represents the point at which half the numbers are above and half are below, providing a clearer picture when the data set has outliers.
Mode Identification
Determining the mode involves finding the number that occurs most frequently in a dataset. The mode is particularly useful for categorical data and can highlight the most common item.
  • List out all numbers in the dataset.

  • Count how many times each number appears.
  • The number that appears the most is the mode.
In our given set of numbers: 1, 1, 2, 3, and 4:
  • Number 1 appears twice, while the others appear only once.

  • This makes 1 the mode of the set.
The mode can help identify peaks in frequency within data, which is valuable in various analytical and reporting tasks.

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Most popular questions from this chapter

Create a sample data set of size \(n=4\) for which the mean \(\bar{x}\), the median \(\tilde{\boldsymbol{x}},\) and the mode are all identical.

State Chebyshev's Theorem.

State the Empirical Rule.

A population data set has mean \(\mu=2\) and standard deviation \(\sigma=1.1\). a. What is the maximum proportion of observations in the data set that can lie outside the interval (-1,3,53)\(?\) b. What can be said about the proportion of observations in the data set that are below -1.3? c. What can be said about the proportion of observations in the data set that are above \(5.3 ?\)

The 54th percentile in a large data set is 98.6 . a. Approximately what percentage of the observations are less than \(98.6 ?\) b. Approximately what percentage of the observations are greater than \(98.6 ?\)
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