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Chapter 3: Problem 62

Using the population formulas, calculate the mean, variance, and standard deviation for the following grouped data. \begin{tabular}{l|ccccc} \hline\(x\) & \(2-4\) & \(5-7\) & \(8-10\) & \(11-13\) & \(14-16\) \\ \hline\(f\) & 5 & 9 & 14 & 7 & 5 \\ \hline \end{tabular}

Short Answer

Expert verified
The mean is approximately 6.15, the variance is 6.375, and the standard deviation is approximately 2.52.

Step by step solution

01

Calculate Mid Points

The midpoint of each interval is calculated by: Midpoint = (Lower limit + Upper limit)/2. Using this, the midpoints of the given group data are: \(3, 6, 9, 12, 15\).
02

Calculate Mean

We calculate the Mean (\(渭\)) as: \(渭 = 危 [Midpoint_i 脳 Frequency_i] / 危 Frequency_i\). Using the formula and the given frequencies, we find that the sum of the products of each midpoint and its frequency is 246. The sum of frequencies is 40. Therefore, \(渭 = 246 / 40 = 6.15\)
03

Calculate Variance

The variance (\(蟽^2\)) is calculated as: \(蟽^2 = [危 (Midpoint_i - 渭)^2 脳 Frequency_i] / 危 Frequency_i\). Firstly, calculate the square of the deviations of each midpoint from the mean, multiply by the frequency, and take the sum of these. In this case, that sum results in 255. Apply the formula and get that \(蟽^2 = 255 / 40 = 6.375 \)
04

Calculate Standard Deviation

Standard Deviation (\(蟽\)) is calculated as: \(蟽 = 鈭歏ariance\). So, \(蟽 = 鈭�6.375 鈮� 2.52\)

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Key Concepts

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

Grouped Data
When working with grouped data, we are dealing with data that's been organized into groups or classes instead of individual data points. This is typical when we have a large dataset and want to simplify it by summarizing it into ranges.
In our exercise, the data is divided into intervals such as 2-4, 5-7, and so on. Each of these intervals is associated with a frequency, which tells us how many data points fall into each range. For example, the interval 2-4 has a frequency of 5.
Grouped data is often used in descriptive statistics as it makes it easier to perform various calculations, such as mean or variance, with large datasets by treating each group as a single data point.
Population Formulas
In statistics, population formulas are used when analyzing the entire dataset or population; this is opposed to a sample. Population formulas provide a way to calculate statistical measures that reflect the entire population's characteristics.
In our exercise, we're using population formulas to calculate measures like mean, variance, and standard deviation. This is suitable when the dataset encompasses every member of the population being studied.
These formulas help account for the variation among data points, giving a true overview of the dataset's dispersion and central tendency.
Mean Calculation
The mean of a dataset is its average, giving us the central value of the data. When calculating the mean for grouped data, we use the midpoints of each interval.
The formula for calculating the mean (\(渭\)) in the context of grouped data is: \[ 渭 = \frac{危 [Midpoint_i 脳 Frequency_i]}{危 Frequency_i} \] Each midpoint represents all values in its interval, multiplied by its frequency to weight its contribution to the overall mean.
  • For example, the midpoint for the interval 2-4 is 3, which represents the entire interval.
  • Then we multiply the midpoint by the interval frequency and sum these results.
  • Finally, divide by the total number of data points (or total frequency) to get the mean.
For our data, this resulted in a mean of 6.15.
Variance Calculation
Variance is a measure of how much the data points in a set differ from the mean. For grouped data, variance can show how spread out the values are around the mean.
The formula for variance (\(蟽^2\)) in grouped data is: \[ 蟽^2 = \frac{危 (Midpoint_i - 渭)^2 脳 Frequency_i}{危 Frequency_i} \] Here are the steps to follow:
  • Calculate the deviation of each midpoint from the mean and square it.
  • Weight these squared deviations using the corresponding frequencies.
  • Sum these weighted squared deviations and divide by the total frequency.
This provides the variance for our group, which shows the degree of spread in the data. In our context, the variance is 6.375, indicating how data is distributed around the mean.
Standard Deviation Calculation
Standard deviation is a measure that summarizes the amount of variation or dispersion in a dataset. It is the square root of the variance, simplifying the interpretation, as it is in the same units as the original data.
For grouped data, we calculate standard deviation by simply taking the square root of the variance: \(蟽 = \sqrt{Variance}\).
In our exercise:
  • The calculated variance was 6.375.
  • Taking the square root gives us the standard deviation.
  • This results in a standard deviation of approximately 2.52, showing us the average distance from the mean.
Understanding standard deviation helps to further understand the consistency and spread of data points within the distribution.

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Most popular questions from this chapter

One disadvantage of the standard deviation as a measure of dispersion is that it is a measure of absolute variability and not of relative variability. Sometimes we may need to compare the variability of two different data sets that have different units of measurement. The coefficient of variation is one such measure. The coefficient of variation, denoted by CV, expresses standard deviation as a percentage of the mean and is computed as follows: For population data: \(\mathrm{CV}=\frac{\sigma}{\mu} \times 100 \%\) For sample data: $$ \mathrm{CV}=\frac{s}{\bar{x}} \times 100 \% $$ The yearly salaries of all employees who work for a company have a mean of \(\$ 62,350\) and a standard deviation of \(\$ 6820\). The years of experience for the same employees have a mean of 15 years and a standard deviation of 2 years. Is the relative variation in the salaries larger or smaller than that in years of experience for these employees?

A large population has a bell-shaped distribution with a mean of 310 and a standard deviation of 37 . Using the empirical rule, find what percentage of the observations fall in the intervals \(\mu \pm 1 \sigma, \mu \pm 2 \sigma\), and \(\mu \pm 3 \sigma\).

A local golf club has men's and women's summer leagues. The following data give the scores for 芒 round of 18 holes of golf for 17 men and 15 women randomly selected from their respective leagues. \begin{tabular}{l|rrrrrrrrr} \hline Men & 87 & 68 & 92 & 79 & 83 & 67 & 71 & 92 & 112 \\ & 75 & 77 & 102 & 79 & 78 & 85 & 75 & 72 & \\ \hline Women & 101 & 100 & 87 & 95 & 98 & 81 & 117 & 107 & 103 \\ & 97 & 90 & 100 & 99 & 94 & 94 & & & \\ \hline \end{tabular} a. Make a box-and-whisker plot for each of the data sets and use them to discuss the similarities and differences between the scores of the men and women golfers. b. Compute the various descriptive measures you have learned for each sample. How do they compare?

The following data give the weights (in pounds) lost by 15 members of a health club at the end of 2 months after joining the club. \(\begin{array}{rrrrrrrr}5 & 10 & 8 & 7 & 25 & 12 & 5 & 14 \\ 11 & 10 & 21 & 9 & 8 & 11 & 18 & \end{array}\) a. Compute the values of the three quartiles and the interquartile range. b. Calculate the (approximate) value of the 8 2nd percentile. c. Find the percentile rank of \(10 .\)

Answer the following questions. a. The total weight of all pieces of luggage loaded onto an airplane is 12,372 pounds, which works out to be an average of \(51.55\) pounds per piece. How many pieces of luggage are on the plane? b. A group of seven friends, having just gotten back a chemistry exam, discuss their scores. Six of the students reveal that they received grades of \(81,75,93,88,82\), and 85, respectively, but the seventh student is reluctant to say what grade she received. After some calculation she announces that the group averaged 81 on the exam. What is her score?
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