/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 In this problem, we explore the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 12

In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set \(2,2,3,6,10\). (a) Compute the mode, median, and mean. (b) Add 5 to each of the data values. Compute the mode, median, and mean. (c) Compare the results of parts \((a)\) and \((b) .\) In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?

Short Answer

Expert verified
Adding a constant increases the mode, median, and mean by that same constant.

Step by step solution

01

Calculating the Mode of the Original Set

The mode is the most frequently occurring value in a data set. For the original set \(2, 2, 3, 6, 10\), the value 2 appears twice while all other values appear only once. Thus, the mode is 2.
02

Calculating the Median of the Original Set

To find the median, arrange the data in ascending order (which is already done) and select the middle value. Since there are five numbers in the set, the median is the third value: 3.
03

Calculating the Mean of the Original Set

The mean is calculated by summing all values in the data set and dividing by the number of values. For the original set:\[\text{Mean} = \frac{2+2+3+6+10}{5} = \frac{23}{5} = 4.6\]
04

Adding 5 to Each Data Value

Add 5 to each number in the original set \(2, 2, 3, 6, 10\) to get the new set: \(7, 7, 8, 11, 15\).
05

Calculating the Mode of the New Set

For the new set \(7, 7, 8, 11, 15\), the value 7 appears twice while all other values appear only once. Thus, the mode is 7.
06

Calculating the Median of the New Set

With the numbers in the new set already arranged, the median is the third value, which is 8.
07

Calculating the Mean of the New Set

The mean is calculated by summing the new set values and dividing by the number of values:\[\text{Mean} = \frac{7+7+8+11+15}{5} = \frac{48}{5} = 9.6\]
08

Comparing the Results

By comparing both sets: - Original mode was 2, new mode is 7 (increase by 5). - Original median was 3, new median is 8 (increase by 5). - Original mean was 4.6, new mean is 9.6 (increase by 5). Adding the constant 5 shifted the mode, median, and mean by the same amount.
09

Generalization

Adding the same number to each data value in a set results in the mode, median, and mean each increasing by that constant. This is because these measures of central tendency shift linearly with the data values.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Mode
Understanding the mode is quite simple. It represents the most frequently occurring number in a set of data. For example, in the dataset \(2, 2, 3, 6, 10\), the number 2 appears more frequently than the others, making 2 the mode. Knowing the mode can be particularly useful when you want to identify which item or value appears the most in a dataset.

When you add a constant to each value in the dataset, such as adding 5 to each number in the original dataset to get \(7, 7, 8, 11, 15\), the mode shifts by the same constant. Originally, the mode was 2; after adding 5 to every value in the dataset, the new mode becomes 7. This shows that the mode changes according to the constant added, maintaining the same frequency pattern as before.
  • Key takeaway: Adding or subtracting a fixed number to all elements in a dataset will change the mode by this constant if the mode repeats with the same frequency.
Median
To find the median, you need to organize your dataset in increasing order and select the middle value. If there's an odd number of data points, it's simple鈥攜ou pick the middle one. In our dataset \(2, 2, 3, 6, 10\), after arranging these values (which are already in order), the third number is 3. This is your median.

Adding a constant shifts all numbers evenly, so when you add 5 to each number, making the dataset \(7, 7, 8, 11, 15\), the median becomes 8. Every element shifts, including the median, by the same constant, preserving the distinct middle position.
  • Key takeaway: Adding or subtracting a fixed number from every point in a data set will translate the median by this constant without affecting the order.
Mean
The mean, or average, is calculated as the sum of all values divided by the number of values. For our initial set \(2, 2, 3, 6, 10\), you calculate it by adding the numbers (23) and dividing by the total quantity (5), resulting in a mean of 4.6.

When a constant is added to each number in your dataset, such as 5 in our example, the overall sum increases proportionally for each added constant. So, the mean moves from 4.6 to 9.6 when 5 is added to each number, reinforcing the idea that every number in the dataset has been altered equally.
  • Key takeaway: The mean reflects even distribution across a dataset. Adding a constant to each number will result in an equivalent shift in the mean by that constant.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set \(5,9,10,11,15\). (a) Use the defining formula, the computation formula, or a calculator to compute \(s\). (b) Multiply each data value by 5 to obtain the new data set \(25,45,50,55,75\). Compute \(s .\) (c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant \(c\) ? (d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be \(s=3.1\) miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations? Given 1 mile \(=1.6\) kilometers, what is the standard deviation in kilometers?

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid E: \(x\) variable \begin{tabular}{lrlllll} \(13.20\) & \(5.60\) & \(19.80\) & \(15.05\) & \(21.40\) & \(17.25\) & \(27.45\) \\ \(16.95\) & \(23.90\) & \(32.40\) & \(40.75\) & \(5.10\) & \(17.75\) & \(28.35\) \\ Grid H: \(y\) variable & & & & & \\ \(11.85\) & \(15.25\) & \(21.30\) & \(17.30\) & \(27.50\) & \(10.35\) & \(14.90\) \\ \(48.70\) & \(25.40\) & \(25.95\) & \(57.60\) & \(34.35\) & \(38.80\) & \(41.00\) \\ \(31.25\) & & & & & & \\ \hline \end{tabular} (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y\), and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(\mathrm{CV}\) s to compare the two grids. If \(s\) represents variability in the sig- -nal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(\mathrm{CV}\) be better, or at least more exciting? Explain.

One indicator of an outlier is that an observation is more than \(2.5\) standard deviations from the mean. Consider the data value \(80 .\) (a) If a data set has mean 70 and standard deviation 5, is 80 a suspect outlier? (b) If a data set has mean 70 and standard deviation 3 , is 80 a suspect outlier?

When data consist of percentages, ratios, growth rates, or other rates of change, the geometric mean is a useful measure of central tendency. For \(n\) data values, Geometric mean \(=\sqrt[n]{\text { product of the } n \text { data values, }}\) assuming all data values are positive To find the average growth factor over 5 years of an investment in a mutual fund with growth rates of \(10 \%\) the first year, \(12 \%\) the second year, \(14.8 \%\) the third year, \(3.8 \%\) the fourth year, and \(6 \%\) the fifth year, take the geometric mean of \(1.10,1.12,1.148,1.038\), and \(1.16 .\) Find the average growth factor of this investment. Note that for the same data, the relationships among the harmonic, geometric, and arithmetic means are harmonic mean \(\leq\) geometric mean \(\leq\) arithmetic mean (Source: Oxford Dictionary of Statistics).

Consider the following ordered data: \(\begin{array}{llllllllll}2 & 5 & 5 & 6 & 7 & 8 & 8 & 9 & 10 & 12\end{array}\) (a) Find the low, \(Q_{1}\), median, \(Q_{3}\) high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.