Presentation on theme: "CHAPTER 1 Exploring Data"— Presentation transcript:
1 CHAPTER 1 Exploring Data 1.3 Describing Quantitative Data with Numbers
2 Histograms on TI-84 Type your data into a list STAT PLOT – use the histogramXlist:Freq: 1Zoom 9Look at the shape. What do you see?Now look at the window and look at the Xscl (width of the bars)
3 Describing Quantitative Data with Numbers CALCULATE measures of center (mean, median).CALCULATE and INTERPRET measures of spread (range, IQR, standard deviation).CHOOSE the most appropriate measure of center and spread in a given setting.IDENTIFY outliers using the 1.5 × IQR rule.MAKE and INTERPRET boxplots of quantitative data.USE appropriate graphs and numerical summaries to compare distributions of quantitative variables.
4 Measuring Center: The Mean and Median Mean: add all of the values (observations) and divide by the total number of observationsMedian: the midpoint of the distribution; half of the observations are smaller and half are largerThere is a quick way to do these calculations on your graphing calculators! (1 – Var Stat)
5 Measuring CenterUse the data below and create a stem and leaf plot. (Make sure to provide a key.) Calculate the mean and median of the commuting times (in minutes) of 20 randomly selected New York workers.103052540201585656045
6 Additional ExampleFor each distribution, describe the shape. Then, find the mean and median and compare them (<, >, =).
7 Measuring Spread: IQR and Five Number Summary A measure of center alone can be misleading. A useful numerical description of a distribution requires both a measure of center and a measure of spread.The 1.5 x IQR Rule for OutliersCall an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile.
8 Find and Interpret the IQR Travel times for 20 New Yorkers:103052540201585656045510152025304045606585510152025304045606585Q1 = 15Median = 22.5Q3= 42.5IQR = Q3 – Q1= 42.5 – 15= 27.5 minutesInterpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes.
9 Recall, this is an outlier by the Construct a BoxplotConsider our New York travel time data:103052540201585656045510152025304045606585Min=5Q1 = 15Median = 22.5Q3= 42.5Max=85Recall, this is an outlier by the1.5 x IQR rule
10 Additional ExampleHere are the amounts of fat in 15 McDonalds beef sandwiches.Make a box-plot with your calculator and sketch it. (Use the trace feature to get the 5 number summary.)Use 1 – Var Stat to find the standard deviation.Calculate and interpret the IQR.Using the 1.5IQR rule, are there any outliers? If so, identify them.
11 Measuring Spread: The Standard Deviation The most common measure of spread looks at how far each observation is from the mean. This measure is called the standard deviation. Consider the following data on the number of pets owned by a group of 9 children.Calculate the mean.Calculate each deviation.deviation = observation – meandeviation: = - 4deviation: = 3= 5
12 Measuring Spread: The Standard Deviation xi(xi-mean)(xi-mean)211 - 5 = -4(-4)2 = 1633 - 5 = -2(-2)2 = 444 - 5 = -1(-1)2 = 155 - 5 = 0(0)2 = 077 - 5 = 2(2)2 = 488 - 5 = 3(3)2 = 999 - 5 = 4(4)2 = 16Sum=?3) Square each deviation.4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance.5) Calculate the square root of the variance…this is the standard deviation.“average” squared deviation = 52/(9-1) = This is the variance.Standard deviation = square root of variance =
13 Measuring Spread: The Standard Deviation The standard deviation sx 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.The average squared distance is called the variance.
14 Choosing Measures of Center and Spread We now have a choice between two descriptions for center and spreadMean and Standard DeviationMedian and Interquartile RangeChoosing Measures of Center and SpreadThe 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!
15 Remember the 4-Step Process? Who Has More Contacts—Males or Females?The following data show the number of contacts that a sample of high school students had in their cell phones. What conclusion can we draw? Give appropriate evidence to support your answer.Males:Females:State: Do males and females differ in the number of contacts they have in their phones?Plan: We will compute numerical summaries, graph parallel boxplots, and compare shape, center, spread, and outliers.
16 Contacts continuedDo: Conclude: In the samples, females typically have more contacts than males, because both the mean and median values are slightly larger. However, the differences are small, so this is not convincing evidence that one gender has more contacts than the other.
17 Data Analysis: Making Sense of Data CALCULATE measures of center (mean, median).CALCULATE and INTERPRET measures of spread (range, IQR, standard deviation).CHOOSE the most appropriate measure of center and spread in a given setting.IDENTIFY outliers using the 1.5 × IQR rule.MAKE and INTERPRET boxplots of quantitative data.USE appropriate graphs and numerical summaries to compare distributions of quantitative variables.