Question

You are given \(n=10\) measurements: 3,5,4,6 , 10,5,6,9,2,8 a. Calculate \(\bar{x}\). b. Find \(m\). c. Find the mode.

Short answer

Answer: a. The mean is 5.8. b. The median is 5. c. The mode is 5 and 6.

Step-by-step solution

Calculate the mean

To find the mean \(\bar{x}\), add all the measurements together and then divide by the total number of measurements (10): \(\bar{x} = \frac{3+5+4+6+10+5+6+9+2+8}{10}\) Now perform the calculations: \(\bar{x} = \frac{58}{10} = 5.8\) The mean is 5.8.

Find the median

To find the median \(m\), first organize the data in ascending order: 2, 3, 4, 5, 5, 6, 6, 8, 9, 10 Since there are an even number of measurements, the median is the average of the two middle values: \(m = \frac{5 + 5}{2} = \frac{10}{2} = 5\) The median is 5.

Find the mode

To find the mode, identify the value that appears most often in the dataset: 2, 3, 4, 5, 5, 6, 6, 8, 9, 10 In this dataset, the numbers 5 and 6 both appear twice, which is more than any other value. Therefore, there are two modes: 5 and 6. a. The mean \(\bar{x}\) is 5.8. b. The median \(m\) is 5. c. The mode is 5 and 6.

Key concepts

Mean

The mean, often referred to as the average, is one of the most basic concepts in descriptive statistics and is used to provide a central value for a dataset.
To calculate the mean, you sum up all the numbers in the dataset and then divide by the total number of observations. In formula terms, this is expressed as \( \bar{x} = \frac{\sum x_i}{n} \), where \( \sum x_i \) is the sum of all observations and \( n \) is the total number of observations.
In the exercise provided, the dataset consists of 10 measurements and the mean (\( \bar{x} \) is found to be 5.8. The mean can provide insights into the general trend of the data, but it can also be influenced by extreme values, which are called outliers. To fully grasp the distribution of data, it is often useful to examine the mean alongside other measures such as the median and mode.

Median

The median is the middle value in a dataset that has been arranged in ascending or descending order. It's a measure of central tendency that indicates the central point of a dataset.
To find the median, you need to list the numbers in order of size. If there’s an odd number of values, the median is the middle one. If there's an even number of values, as in the exercise, you calculate the average of the two middle numbers. The formula for the median when the dataset size \( n \) is even is given by \( m = \frac{( n/2 )^\text{th} term + ( (n/2) + 1 )^\text{th} term}{2} \).
The median for the given measurements is 5, which is a robust indicator of central tendency because it is not affected by outliers or skewed data, unlike the mean.

Mode

The mode is the most frequently occurring value in a set of data. Unlike the mean and the median, the mode can be used for both numerical and nominal (non-numerical) data.
In certain datasets, there can be more than one mode if multiple values appear with the same highest frequency; such a dataset is referred to as 'bimodal' or 'multimodal' depending on the number of modes. In the example exercise, the dataset has two modes, 5 and 6, making it bimodal. Identifying the mode can be particularly useful for categorical data where you want to know which is the most common category. Although the mode is easy to compute, it may not provide a good measure of central tendency if the dataset has no repeats or if all values have the same frequency.

Data Analysis

Data analysis involves inspecting, cleansing, transforming, and modeling data to discover useful information, inform conclusions, and support decision-making.
In descriptive statistics, data analysis includes using summary measures like mean, median, and mode to represent features of the data. After calculating these measures, you can gain a better understanding of the data distribution, its central tendency, and the variability among data points.
Using the exercise as a case study, after finding the mean (5.8), median (5), and mode(s) (5 and 6), we can say that the data points generally center around 5 to 6. With such knowledge, one can perform further data analysis tasks, like creating graphical representations, to enhance understanding and effectively communicate the results.