Objective To understand measures of central tendency and use them to analyze data.

Measures of Central Tendency Mean, Median and Mode Mean – the average Sum of the data items Total number of data items Used to describe the middle of a set of data that does not have outliers (data values that are much higher or lower than other values in the set.)

Find the Mean Q: 4, 5, 8, 7 A: 6 Median: 6 Q: 4, 5, 8, 1000 A: 254.25

Median The middle value in a set of data where the numbers are arranged in order. Used to describe the middle of a set of data that does have outliers. If the data has an even number of items the median is the average of the middle two numbers.

Median Find the Median 4 5 6 6 7 8 9 10 12 5 6 6 7 8 9 10 12 5 6 6 7 8 9 10 100,000

Mode The data item that occurs the most times Can be used when data is not numeric Can have one, two or more modes. Used to choose the most popular outcome.

Mode Most Common Outcome Male Female

Measures of Central Tendency Mode The most common observation in a group of scores. Distributions can be unimodal, bimodal, or multimodal. If the data is categorical (measured on the nominal scale) then only the mode can be calculated. The most frequently occurring score (mode) is Vanilla. Flavor f Vanilla 28 Chocolate 22 Strawberry 15 Neapolitan 8 Butter Pecan 12 Rocky Road 9 Fudge Ripple 6

Range The difference between the least and greatest data values. Find the range of: 2, 34, 55, 22, 4, 7, 84, 55, 77

Summarizing Distributions Two key characteristics of a frequency distribution are especially important when summarizing data or when making a prediction from one set of results to another: Central Tendency What is in the “Middle”? What is most common? What would we use to predict? Dispersion How Spread out is the distribution? What Shape is it?

Measures of Variability Central Tendency doesn’t tell us everything Dispersion/Deviation/Spread tells us a lot about how a variable is distributed. We are most interested in Standard Deviations (σ)

Standard Deviation A measure of dispersion that describes the typical difference or deviation between the mean and a data value. Subtract the mean from each data value and square this difference. Do this for each data value and add the answers together. Divide the sum by the number of data items then take the square root.

Find the standard deviation for the following test scores 98, 72, 55, 88, 69, 92, 77, 89, 94, 70

Line Plot Used to show frequency.

Tally Chart Used to show frequency.

Stem and Leaf Plot Used to organize data. Easy to see the mode!

Practice! Use the following data to make a stem and leaf plot. Find the mean, median, mode and range of the data. 18, 35, 28, 15, 36, 72, 14, 55, 62, 45, 80, 9, 72, 66, 28, 20, 51, 44, 28 Mean 40.95 Median 36 Mode 28 Range 71

1-4 Bar Graphs and Histograms Course 2 1-4 Bar Graphs and Histograms A histogram is a bar graph that shows the frequency of data within equal intervals. There is no space between the bars in a histogram.

The histogram is a tool for presenting the distribution of a numerical variable in graphical form. For example, suppose the following data is the number of hours worked in a week by a group of nurses: 42 47 43 26 30 42 28 42 50 39 38 35 37 48 39 36 45 41 72 53 43 37 42 48 40 35 39 30 47 38

These data are displayed in the following histogram: The data values are grouped in intervals of width five hours. The first interval includes the values from 25 to less than 30 hours. The second interval includes values from 30 to less than 35 and so on. The intervals are shown on the horizontal axis. 26 28 30 35 36 37 38 39 40 41 42 43 45 47 48 50 53 72 The vertical axis is frequency. So, for example, there are two nurses who worked from 25 to less than 30 hours that week.

The choice of interval width will affect the appearance of the histogram. And here it is again, to the right, presented in a histogram of interval width 2. To the right is the same data presented in a histogram of interval width 10.

Additional Example 3: Making a Histogram Course 2 1-4 Bar Graphs and Histograms Additional Example 3: Making a Histogram The table below shows the number of hours students watch TV in one week. Make a histogram of the data. Step 1: Make a frequency table of the data. Be sure to use equal intervals. 6 /// 7 //// //// 8 /// 9 //// 1 // 2 //// 3 //// //// 4 //// / 5 //// /// Number of Hours of TV Frequency Number of Hours of TV 1–3 15 4–6 17 7–9 17

Additional Example 3 Continued Course 2 1-4 Bar Graphs and Histograms Additional Example 3 Continued Step 2: Choose an appropriate scale and interval for the vertical axis. The greatest value on the scale should be at least as great as the greatest frequency. 20 16 12 8 4 1–3 Frequency Number of Hours of TV 15 4–6 17 7–9

Additional Example 3 Continued Course 2 1-4 Bar Graphs and Histograms Additional Example 3 Continued Step 3: Draw a bar graph for each interval. The height of the bar is the frequency for that interval. Bars must touch but not overlap. 20 16 12 8 4 1–3 Frequency Number of Hours of TV 15 4–6 17 7–9

1-4 Bar Graphs and Histograms Additional Example 3 Continued Course 2 1-4 Bar Graphs and Histograms Additional Example 3 Continued Step 4: Label the axes and give the graph a title. Hours of Television Watched 20 16 12 8 4 1–3 Frequency Number of Hours of TV 15 4–6 17 7–9 Frequency 1–3 4–6 7–9 Hours

1-4 Bar Graphs and Histograms Try This: Example 3 1–3 12 4–6 18 7–9 24 Course 2 1-4 Bar Graphs and Histograms Try This: Example 3 The table below shows the number of hats a group of students own. Make a histogram of the data. Step 1: Make a frequency table of the data. Be sure to use equal intervals. 1 // 2 //// 3 //// / 4 //// / 5 //// /// 6 //// 7 //// / 8 //// //// 9 //// //// Number of Hats Owned Frequency Frequency Number of Hats Owned 1–3 12 4–6 18 7–9 24

1-4 Bar Graphs and Histograms Try This: Example 3 1–3 12 4–6 18 7–9 24 Course 2 1-4 Bar Graphs and Histograms Try This: Example 3 Step 2: Choose an appropriate scale and interval for the vertical axis. The greatest value on the scale should be at least as great as the greatest frequency. 30 25 20 15 10 5 Frequency Number of Hats Owned 1–3 12 4–6 18 7–9 24

1-4 Bar Graphs and Histograms Try This: Example 3 1–3 12 4–6 18 7–9 24 Course 2 1-4 Bar Graphs and Histograms Try This: Example 3 Step 3: Draw a bar graph for each interval. The height of the bar is the frequency for that interval. Bars must touch but not overlap. 30 25 20 15 10 5 Frequency Number of Hats Owned 1–3 12 4–6 18 7–9 24

1-4 Bar Graphs and Histograms Try This: Example 3 Number of Hats Owned Course 2 1-4 Bar Graphs and Histograms Try This: Example 3 Number of Hats Owned Step 4: Label the axes and give the graph a title. 30 25 20 15 10 5 Frequency Frequency Number of Hats Owned 1–3 12 4–6 18 7–9 24 1–3 4–6 7–9 Number of Hats

1-4 Bar Graphs and Histograms Lesson Quiz: Part 1 Course 2 1-4 Bar Graphs and Histograms Lesson Quiz: Part 1 1. The list shows the number of laps students ran one day. Make a histogram of the data. 4, 7, 9, 12, 3, 6, 10, 15, 12, 5, 18, 2, 5, 10, 7, 12, 11, 15 Number of Students Number of Laps Run 10–14 0–4 5–9 8 6 4 2 15–19 Number of Laps

Normally Distributed Curve

Characteristics of the Normal Distribution It is symmetrical -- Half the cases are to one side of the center; the other half is on the other side. The distribution is single peaked Most of the cases will fall in the center portion of the curve and as values of the variable become more extreme they become less frequent, with “outliers” at each of the “tails” of the distribution few in number. The Mean, Median, and Mode are the same. Percentage of cases in any range of the curve can be calculated.

Skewed Distributions

The total area under the normal curve is 1. About 68% of the area lies within 1 standard deviation of the mean. About 95% of the area lies within 2 standard deviations of the mean. About 99.7 % of the area lies within 3 standard deviations of the mean.

Why can’t the mean tell us everything? Mean describes Central Tendency, what the average outcome is. We also want to know something about how accurate the mean is when making predictions. The question becomes how good a representation of the distribution is the mean? How good is the mean as a description of central tendency -- or how good is the mean as a predictor? Answer -- it depends on the shape of the distribution. Is the distribution normal or skewed?

Dispersion Once you determine that the variable of interest is normally distributed, the next question to be asked is about its dispersion: how spread out are the scores around the mean. Dispersion is a key concept in statistical thinking. How much do the scores deviate around the Mean? The more “bunched up” around the mean the better your ability to make accurate predictions.

How well does the mean represent the scores in a distribution How well does the mean represent the scores in a distribution? The logic here is to determine how much spread is in the scores. How much do the scores "deviate" from the mean? Think of the mean as the true score or as your best guess. If every X were very close to the Mean, the mean would be a very good predictor. If the distribution is very sharply peaked then the mean is a good measure of central tendency and if you were to use the mean to make predictions you would be right or close much of the time.