1. + Chapter 1: Exploring Data Section 1.3 Describing Quantitative Data with Numbers The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE

2. + Describing Quantitative Data Measuring Spread: The Standard Deviation Definition: 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.

3. + Describing Quantitative Data Measuring Spread: The Standard Deviation Consider the following data on the number of pets owned bya group of 9 children. 1)Calculate the mean. 2)Calculate each deviation. deviation = observation – mean = 5 deviation: 1 - 5 = -4 deviation: 8 - 5 = 3

4. + Describing Quantitative Data Measuring Spread: The Standard Deviation xixi (x i -mean)(x i -mean) 2 11 - 5 = -4(-4) 2 = 16 33 - 5 = -2(-2) 2 = 4 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 55 - 5 = 0(0) 2 = 0 77 - 5 = 2(2) 2 = 4 88 - 5 = 3(3) 2 = 9 99 - 5 = 4(4) 2 = 16 Sum=? 3) Square each deviation. 4) Sum the squared deviations. Then, divide by (n-1)…this is called the variance (the “average” squared deviation). 5) Take the square root of the variance…this is the standard deviation. “average” squared deviation = 52/(9-1) = 6.5 This is the variance. Standard deviation = square root of variance =

5. + Choosing Measures of Center and Spread We now have a choice between two descriptions for centerand spread Mean and Standard Deviation Median and Interquartile Range The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! Choosing Measures of Center and Spread Describing Quantitative Data

6. + Transformations Adding, subtracting, multiplying and dividing all the numbers in a data set is called “transforming” the data. ONE OF THE CENTRAL CONCEPTS FROM THIS CHAPTER is knowing how transformations affect measures of center and spread. I WILL TEST ON THIS. OFTEN. A LOT. MeanStandard deviation Adding or subtractingMean gets added or subtracted same amount Standard deviation doesn’t change Multiplying or dividingMean gets multiplied or divided by same amount Standard deviation gets multiplied or divided by same amount Think about this like a curve on a test. If I add 5 points to everyone’s test, how will the class average change? Think about this like a curve on a test. If I add 5 points to everyone’s test, how will the spread of the grades change?

7. + Example question A researcher wishes to calculate the average height of patients suffering from a particular disease. From patient records, the mean was computed as 156 cm, and the standard deviation as 5 cm. Further investigation reveals that the scale was misaligned, and that all readings are 2cm too large, for example a patient whose height was really 180 cm was measured as 182 cm. Furthermore the researcher would like to work with statistics based on meters. The correct mean and standard deviation are: A) 1.56 m, 0.05 mB) 1.54 m, 0.05 m C) 1.56 m, 0.03 mD) 1.58 m, 0.05 m E) 1.58 m, 0.07 m

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