Math 310 Section 8.1 & 8.2 Statistics
Centers and Spread A goal in statistics is to determine how data is centered and spread. There are many different ways to measure these two aspects.
Measures of Center • Mean • Median • Mode
Measures of Spread • Range • Interquartile range (IQR) • Variance • Standard Deviation
Mean Def The arithmetic mean of the numbers x1, x2, …, xn, denoted (x-bar) is given by: (x-bar) = x1 + x2 + ... + xn n
Ex. Find the mean of the data : {1, 2, 3, 4, 5} = (1 + 2 + 3 + 4 + 5)/5 = 15/5 = 1/3
Median The median is the middle score when data is arranged in order. If there are an odd number of pieces of data, then the median will be one of the pieces of data. If there are an even number of pieces of data, then the median is the mean of the two middle scores.
Ex. Find the median of {2, 6, 9 , 3, 5, 1, 12} Ordering the data: 1, 2, 3, 5, 6, 9, 12 And the median is 5. Find the median of {74, 21, 49, 50} Ordering the data: 21, 49, 50, 74 And the median is (49 + 50)/2 = 49.5
Mode The modeis the number that appears most often in a set of data. It is possible to have more than one mode. If there are two modes we say that that the data is bimodal. If no number appears more than once you can say there is no mode, or that every number is the mode.
Ex. Find the mode of the following data: {8, 3, 4, 3, 0, 8, 8, 9, 12} Mode: 8
Range The range is the difference of the highest and lowest scores in a set of data.
Ex. Find the range of {12, 14, 549, 34, 2343} Range: 2343 – 12 = 2331
Interquartile Range Using the median three times we can divide a set of data into quarters. The lower quartile or Q1 is the number dividing the first and second quarters. The upper quartile or Q3 is the number dividing the second and third quarters. The IQR is the difference of Q3and Q1.
Ex. Take the set of data {20, 25, 40, 50, 50, 60, 70, 75, 80, 80, 90, 100, 100}. What is the lower quartile, the upper quartile, and the IQR? Q1 = 45, Q2 = 85 So IQR = 40
Variance To find the variance: • Find the mean of the numbers. • Subtract the mean from each number • Square each difference found in step 2. • Find the sum of the squares in step 3. • Divide by n to obtain the variance v.
Ex. Find the variance of {2, 4, 7, 2, 0, 4, 6, 8, 3} Mean: 4 v = [(4-2)2 + (4-4)2 + … + (4-3)2]/9 = 6
Standard Deviation To find the standard deviation take the square root of the variance.
Ways to view data • Pictographs • Bar graph • Histogram • Dot plot • Stem-and-leaf plot • Frequency Table • Scatter plot • Line graph • Circle graph • Box Plot
When to use what? For a list of when to use what kind of representation refer to pg 516 of our text.
