2.5 Measures of Center of the Data
- Page ID
- 36467
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives
In this section, you will:
• Measure the centers of data, including mean, median, and mode.
The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median.
Mean and Median
Mean: add all data, divide by the total number of values.
Median: middle value when the data are placed in order. If there is no middle value, find the mean of the two middle values.
• The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers.
• You can quickly find the location of the median by using the expression (n + 1)/2.
Example 1
Consider the following data: 19; 18; 18; 25; 24; 32; 45; 29; 17; 18; 53; 30; 20; 21.
Find mean.
Find median.
Using the graphing calculator to find the mean and median.
• Clear list L1. Press STAT 4: ClrList. Enter 2nd 1 for list L1. Press ENTER.
• Enter data into the list editor. Press STAT 1: EDIT.
• Put the data values into list L1.
• Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and then Calculate.
• Press the down and up arrow keys to scroll.
Example 2
AIDS data indicating the number of months a patient with AIDS lives after taking a new antibody drug are as follows (smallest to largest): 3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47.
Calculate the mean and the median.
Mode
Another measure of the center is the mode. The is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.
Example 3
Statistics exam scores for 20 students are as follows: 50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72;
76; 78; 81; 83; 84; 84; 84; 90; 93.
Find the mode.
Calculating the Mean of Frequency Distribution
When only grouped data is available, you do not know the individual data values (we only know intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table.
o f = the frequency of the interval
o m = the midpoint of the interval
Example 4
A frequency table displaying Professor Blount’s last statistic test is shown. Find the best estimate of the class mean.
| Grade Interval | Number of Students |
| 50.5-56.4 | 1 |
| 56.5-62.4 | 0 |
| 62.5-68.4 | 4 |
| 68.5-74.4 | 4 |
| 74.5-80.4 | 2 |
| 80.5-86.4 | 2 |
| 86.5-92.4 | 4 |
| 92.5-98.4 | 1 |
Using the graphing calculator to find the mean of grouped frequency tables.
• Clear list L1. Pres STAT 4:ClrList. Enter 2nd 1 for list L1. Press ENTER.
• Enter data into the list editor. Press STAT 1:EDIT.
• Put the midpoint values into list L1.
• Put the frequency values into list L2.
• Press STAT and arrow to CALC. Press 1:1-VarStats.
• List: Press 2nd 1 for L1.
• FreqList: Press 2nd 2 for L2. and then Calculate.
• Press the down and up arrow keys to scroll.
For more information and examples see online textbook OpenStax Introductory Statistics pages 100- 106.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution License 4.0 license