: Shape, Center, and Spread Opening Exercise Distributions - Data are often summarized by graphs. We often refer to the group of data presented in the graph as a distribution. Below are examples of the two types of graphs we will be using most. DOT PLOTS: A plot of each data value on a scale or number line. HISTOGRAMS: A graph that groups the data based on intervals and represents the amount of data in each interval by the height of a bar. 1. Use the data below to create a dot plot. Then comment on its shape. A sample was taken of people who were attending a birthday party. The ages of the ten people in the sample are as follows:,,,,,,,,, 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Unit 12: Normal Distributions S.139
Once we have the graph of a distribution, we often want to describe the graph. We generally describe a distribution by commenting on its shape, center and spread. To describe the shape, we will generally use symmetric, skewed left, or skewed right Symmetric Skewed Right Skewed Left 2. Describe the shape of the data from Exercise 1. Center: To describe the center, we calculate the mean, median, or mode. (Though we will rarely use the mode). verbally x bar verbally mu Notation: For the mean of a sample, we write. For the mean of an entire population, we write. 3. A. Calculate the mean, median, and mode of the data from Exercise 1. Mean: Median: Mode: = B. How do the mean and the median in our data set compare? Unit 12: Normal Distributions S.140
Spread: To describe the spread, we will calculate the standard deviation or range. Range is the difference between the largest and smallest values in the data set. range = maximum value minimum value 4. In this data set, the range is. In Algebra 1 you studied standard deviation. What is a standard deviation? We use standard deviation to measure the variability among the numbers in the data set. The larger the standard deviation, the more variability there is. 5. Interpretation of Standard Deviation: The standard deviation of a set of numbers tells us: the distance between a number in set and the. verbally sigma Notation: The notation for the standard deviation of a sample is s. The notation for the standard deviation of a population is. verbally sigma For this lesson we will rely on technology to calculate standard deviation. We will review how to calculate it in a later lesson. You will need: Chromebook, Using Desmos for Statistics handout 6. Use the direction on the handout to input each data set. Determine the mean, median and standard deviation for each set of data. Then interpret what the standard deviation is telling us about the data. Mean Median Standard Deviation Data Set 1 Data Set 2 Exercise 1 Data 6, 7, 10, 11, 26 8, 10, 14, 16 4, 4, 6, 6, 6, 7, 7, 8, 15, 17 7. Let s look back at our age data from Exercise 1. How can we interpret the standard deviation? A typical age in this group of people is years away from the mean. Unit 12: Normal Distributions S.141
Which statistics do we use? If the distribution is symmetric, we will generally use the mean as our measure of center and the standard deviation as our measure of spread. If the distribution is skewed, we frequently use the median as our measure of center and the range as our measure of spread. 8. A local baseball club, the Manatees, has twelve players. The batting averages for those players are as follows: 0.255, 0.260, 0.265, 0.275, 0.275, 0.275, 0.280, 0.280, 0.280, 0.280, 0.285, 0.285 A. Create a dot plot for the batting averages. 0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 B. Describe the shape of the dot plot. C. Find the mean and median of the data set. Which is larger? D. Find the standard deviation of the batting averages using technology. E. Interpret the meaning of the standard deviation. Unit 12: Normal Distributions S.142
9. A paleontologist studies prehistoric life and sometimes works with dinosaur fossils. The histogram below shows the distribution of heights (rounded to the nearest centimeter) of 660 procompsognathids, otherwise known as compys, based on fossil findings. A. Describe the shape of the distribution B. The mean of the distribution of compy heights is 33.5 cm; the median is also 33.5.cm. What is the relationship between the mean and median here? C. How could we estimate the mode of this distribution? D. The standard deviation of the distribution is 2.56 cm. Interpret the value in this context. Unit 12: Normal Distributions S.143
Drawing Conclusions The shape of a distribution tells us the relationship between its mean and median. Symmetric Example: Skewed Right Example: Skewed Left Example: 10. The mean is the median. This is similar to the dot plot in Exercise. 11. The mean is the median. This is similar to the dot plot in Exercise. 12. The mean is the median. This is similar to the dot plot in Exercise. Did you know? Skewed distributions tend to have larger standard deviations than symmetric distributions. Distributions with a larger range also tend to have a higher standard deviation. The mean is generally pulled in the direction of extreme values (outliers) so it is affected by skew. Unit 12: Normal Distributions S.144
The histogram below shows the distribution of the greatest drop (in feet) for 55 major roller coasters in the United States. 13. A. How would you describe the shape of the distribution? B. The median is approximately 120 ft. Is the mean of the maximum drop distribution closest to 90, 120, or 135 feet? Explain your answer. C. Would you use the mean or the median as the measure of center for this distribution? Why? D. Which measure of spread would you use for this distribution? Explain, then calculate it. Unit 12: Normal Distributions S.145
You will need: 8 Summary Statistics cards and 8 Histogram cards 14. Distribution Matching Activity Match each histogram card with the summary statistics card that most closely describes it. Statistics Summary Number 1 2 3 4 5 6 7 8 Histogram Number 15. Discussion What strategies did you use to match the statistics to the histograms? Lesson Summary The shape of a distribution is often described as symmetric, skewed left, or skewed right. We usually describe a distribution by commenting on its shape, center, and spread. When the distribution is symmetric we will usually use the mean to describe the center and the standard deviation to describe the spread. When the distribution has extreme skew, we will usually choose the median for the center and the range for the spread. Unit 12: Normal Distributions S.146