1.3: Describing Quantitative Data with Numbers

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1 1.3: Describing Quantitative Data with Numbers Section 1.3 Describing Quantitative Data with Numbers After this section, you should be able to MEASURE center with the mean and median MEASURE spread with standard deviation and interquartile range IDENTIFY outliers CONSTRUCT a boxplot using the five number summary CALCULATE numerical summaries with technology Measuring Center: The Mean x To find the mean (pronounced x bar ) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x, x 3,, x n, their mean is: x sum of observations x 1 x... x n n n Compact Notation: x i x n Measuring Center: The Median The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1)Arrange all observations from smallest to largest. )If the number of observations n is odd, the median M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list. Comparing the Mean and the Median The mean and median measure center in different ways, and both are useful. Mean: average value Median: typical value Relationship between Mean & Median: The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median.

2 Why is the mean more affected by the presence of outliers than the median? 1.3 Activity Drive Time at ATM You need to know how to calculate mean and standard deviation with your calculator. We are working in groups of 8 TWO Tables. Standard Deviation A relatively low standard deviation value indicates that the data points tend to be very close to the mean. A relatively high standard deviation value indicates that the data points are spread out over a large range of values. Standard Deviation Formula The standard deviation s x measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance. variance = ( x1 x) ( x x)... ( xn x) sx n 1 1 n 1 ( x x) i standard deviation = s x 1 (x i x ) n 1 Two Extreme Examples: In dataset #1, we have five people that report eating 4 pieces of cake and five people that report eating 6 pieces of cake, for a mean of 5 pieces of cake ([ ]/10=5). Mean =5; Variance = 1 In dataset #, we have five people that report eating 0 piece of cake and five people that report eating 10 pieces of cake, for a mean of 5 pieces of cake ([ ]/10=5). Mean = 5; Variance = 5 Below are dotplots of three different distributions, A, B, and C. Which one has the largest standard deviation? Justify your answer.

3 FRQ 018 #5 Interquartile Range (IQR) Interquartile Range (IQR) To calculate: 1)Arrange the observations in increasing order and locate the median M. )The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3)The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: Find and Interpret the IQR Travel times to work for 0 randomly selected New Yorkers IQR = Q 3 Q 1 Find and Interpret the IQR Travel times to work for 0 randomly selected New Yorkers Q 1 = 15 M =.5 Q 3 = 4.5 IQR = Q 3 Q 1 = = 7.5 minutes Identifying Outliers In addition to serving as a measure of spread, the interquartile range (IQR) is used as part of a rule of thumb for identifying outliers. 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile. Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 7.5 minutes.

4 In the New York travel time data, we found Q 1 =15 minutes, Q 3 =4.5 minutes, and IQR=7.5 minutes. Calculate the outlier cutoffs using the IQR rule. In the New York travel time data, we found Q 1 =15 minutes, Q 3 =4.5 minutes, and IQR=7.5 minutes. Calculate the outlier cutoffs using the IQR rule. For these data, 1.5 x IQR = 1.5(7.5) = 41.5 Q x IQR = = 6.5 Q x IQR = = Any travel time shorter than 6.5 minutes or longer than minutes is considered an outlier. The Five Number Summary The five number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q 1 M Q 3 Maximum TI Nspire: 5 Number Summary 1. Select Lists & Spreadsheet (bottom of home screen). Type the values into list1. 3. With your cursor on the values, press menu 4. Select 4: Statistics, then 1: Stat Calculations, press enter. 5. Select 1: One Variable Stats TI Nspire: 5 Number Summary 6. Set screen to: and then press enter. 7. Scroll down to see the 5 number summary.

5 Boxplots (Box and Whisker Plots) Draw and label a number line that includes the range of the distribution. Construct a Boxplot Using our NY travel times data. Construct a boxplot. Draw a central box from Q 1 to Q 3. Note the median M inside the box. Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers TravelTime Construct a Boxplot Using our NY travel times data. Construct a boxplot. Choosing Best Measures of Center & Spread Symmetric Distribution Skewed Distribution Min=5 Q 1 = 15 M =.5 Q 3 = 4.5 Max=85 Recall, this is an outlier by the 1.5 x IQR rule Best Measure of Center Best Measure of Spread TravelTime Desmos Activity Go to student.desmos.com and type in: TCKZ6U FYI: Why n 1?! Applet: sticsapplets/n 1.html OR CLICK: ULATECODE=TCKZ6U Proof

6 How to Calculate Standard Deviation by Hand 1. Calculate mean.. Calculate each deviation. Subtract your mean score from every actual (observed) score. 3. Square each deviation. 4. Find the average squared deviation by calculating the sum of the squared deviations divided by (n 1). 5. Finally, calculate the square root of the answer in step #4 Calculate the standard deviation NumberOfPets 1) Calculate the mean. Step 1: 5 ) Calculate each deviation. deviation = observation mean deviation: 8-5 = 3 deviation: 1-5 = NumberOfPets x = 5 x i Sum= (x i mean) 3) Square each deviation. Step 3: See Table 4) Find the average squared deviation by calculating the sum of the squared deviations divided by (n 1). Step 4: Average squared deviation = 5/(9 1) = 6.5 Variance = 6.5 x i (x i mean) (x i mean) = = = = = = = = = 4 Sum= Sum= 5) Calculate the square root of the variance this is the standard deviation. Step 5: Square root of variance Standard Deviation =.55 x i (x i mean) (x i mean) = 4 ( 4) = = ( ) = = 1 ( 1) = = 1 ( 1) = = 1 ( 1) = = 0 (0) = = () = = 3 (3) = = 4 (4) = 16 Sum=? Sum=?