Measures of Central Tendency

Objective To learn how to find measures of central tendency in a set of raw data.

Relevance To be able to calculate the most appropriate measure of center after analyzing the context of a study that might or might not contain extreme values.

There are 4 values that are considered measures of the center. 1. Mean Central Values – Many times one number is used to describe the entire sample or population. Such a number is called an average. There are many ways to compute an average. There are 4 values that are considered measures of the center. 1. Mean 2. Median 3. Mode 4. Midrange

Arrays Mean – the arithmetic average with which you are the most familiar. Mean:

Sample and Population Symbols As we progress in this course there will be different symbols that represent the same thing. The only difference is that one comes from a sample and one comes from a population.

Symbols for Mean Sample Mean: Population Mean:

Rounding Rule Round answers to one decimal place more than the number of decimal places in the original data. Example: 2, 3, 4, 5, 6, 8 A Sample answer would be 4.1

Example Find the mean of the array. 4, 3, 8, 9, 1, 7, 12

Example……. Find the mean of the following numbers. 23, 25, 26, 29, 39, 42, 50

Example 2 – Use GDC Find the mean of the array. 2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6 Use your lists on the calculator and follow the steps.

Stat, Edit – input list

Stat, Calc, One-Var Stats, L1

Or…..(I like this way better!) Home: 2nd Stat Math 3: Mean (L#)

Rounding The mean (x-bar) is 5.77. We used 2 decimal places because our original data had 1 decimal place.

Median Median – the middle number in an ordered set of numbers. Divides the data into two equal parts. Odd # in set: falls exactly on the middle number. Even # in set: falls in between the two middle values in the set; find the average of the two middle values.

Example Find the median. A. 2, 3, 4, 7, 8 - the median is 4. B. 6, 7, 8, 9, 9, 10 median = (8+9)/2 = 8.5.

Ex 2 – Use Calculator Input data into L1.

Run “Stat, Calc, One-Variable Stats, L1” Cursor all the way down to find “med”

Or…….from the home screen 2nd Stat Math 4: Median(L#)

Mode The number that occurs most often. Suggestion: Sort the numbers in L1 to make it easier to see the grouping of the numbers. You can have a single number for the mode, no mode, or more than one number.

Example Find the mode. 1, 2, 1, 2, 2, 2, 1, 3, 3 Put numbers in L1 and sort to see the groupings easier.

The mode is 2.

Ex 2 Find the mode. A. 0, 1, 2, 3, 4 - no mode B. 4, 4, 6, 7, 8, 9, 6, 9 - 4 ,6, and 9

Midrange The number exactly midway between the lowest value and highest value of the data set. It is found by averaging the low and high numbers.

Example Find the midrange of the set. 3, 3, 5, 6, 8

Trimmed Mean

Trimmed Mean We have seen 4 different averages: the mean, median, mode, and midrange. For later work, the mean is the most important. However, a disadvantage of the mean is that it can be affected by extremely high or low values. One way to make the mean more resistant to exceptional values and still sensitive to specific data values is to do a trimmed mean. Usually a 5% trimmed mean is used.

How to Compute a 5% Trimmed Mean Order the data from smallest to largest. Delete the bottom 5% of the data and the top 5% of the data. (NOTE: If 5% is a decimal round to the nearest integer) Compute the mean of the remaining 90%.

Example a) Compute the mean for the entire sample. b) Compute a 5% trimmed mean.

Example c) Compute the median for the entire sample. d) Compute a 5% trimmed median. The median is still 32.5. e) Is the trimmed mean or the original mean closer to the median? Trimmed Mean

Weighted Average

Sometimes we wish to average numbers, but we want to assign more importance, or weight, to some of the numbers. The average you need is the weighted average.

Formula for Weighted Average

Example: You will earn an A! Suppose your midterm test score is 83 and your final exam score is 95. Using weights of 40% for the midterm and 60% for the final exam, compute The weighted average of your scores. If the minimum average for an A is 90, will you earn an A? You will earn an A!

Measures of Dispersion…..Arrays

Dispersion The measure of the spread or variability No Variability – No Dispersion

Measures of Variation There are 3 values used to measure the amount of dispersion or variation. (The spread of the group) 1. Range 2. Variance 3. Standard Deviation

Why is it Important? You want to choose the best brand of paint for your house. You are interested in how long the paint lasts before it fades and you must repaint. The choices are narrowed down to 2 different paints. The results are shown in the chart. Which paint would you choose?

The chart indicates the number of months a paint lasts before fading. Paint A Paint B 10 35 60 45 50 30 40 20 25 210

Does the Average Help? Paint A: Avg = 210/6 = 35 months Paint B: Avg = 210/6 = 35 months They both last 35 months before fading. No help in deciding which to buy.

Consider the Spread Paint A: Spread = 60 – 10 = 50 months Paint B: Spread = 45 – 25 = 20 months Paint B has a smaller variance which means that it performs more consistently. Choose paint B.

Range The range is the difference between the lowest value in the set and the highest value in the set. Range = High # - Low #

Example Find the range of the data set. 40, 30, 15, 2, 100, 37, 24, 99 Range = 100 – 2 = 98

Deviation from the Mean A deviation from the mean, x – x bar, is the difference between the value of x and the mean x bar. We base our formulas for variance and standard deviation on the amount that they deviate from the mean. We’ll use a shortcut formula – not in book.

Variance (Array) Variance Formula

Standard Deviation The standard deviation is the square root of the variance.

Example – Using Formula Find the variance. 6, 3, 8, 5, 3

Find the standard deviation The standard deviation is the square root of the variance.

Same Example – Use Calculator Put numbers in L1.

Run “Stat, Calc, One-Variable Stats, L1” and read the numbers Run “Stat, Calc, One-Variable Stats, L1” and read the numbers. Remember you have to square the standard deviation to get variance. Square this number to get the variance!

Or….from the home screen 2nd Stat Math 7:stdDev(L1) Enter

Variance – Using Formula Square the ENTIRE number for the standard deviation not the rounded version you gave for your answer.

Variance using GDC 2nd Stat Math 8: Variance (L1)