Measures of Center & Spread

Measures of Center

Example: What if another 5 is added to the data? Now find just the mode First rearrange the data from least to greatest: Mode: None

Measures of Center

Measures of Center How can we make the calculator do this for us? “Stat” “Edit” Enter data into L 1 “Stat” “Calc” “1-Var Stats” List: L 1 (or applicable list) “Calculate”

Measures of Center Example: A teenager recoded the time (in minutes per day) he spent playing Candy Crush over a 2 week period: Using your GDC, determine the mean and median daily time spent playing this game. Mean: 98.4 minutes; median: minutes Be sure to include units!

Measures of Center Outliers: data values that are either much larger or much smaller than the general body of data. Resistance: the sensitivity of an estimator to extreme observations. Estimators that do not change much with the addition or deletion of extreme observations are said to be resistant.

Effects of Outliers Let’s look at an example of each measure of center with and without an outlier. 1.Mean? Example A: 1, 2, 3, 4, 5 Example B: 1, 2, 3, 4, Median? Example A: 1, 2, 3, 4, 5 Example B: 1, 2, 3, 4, Mode? Example A: 1, 2, 3, 3, 4, 5 Example B: 1, 2, 3, 3, 4,

Effects of Outliers The mean is a “follower” – it will go towards the skewness! Skewed left: Skewed right: Mean Median

Effects of Outliers And if the data is approximately symmetrical? Mean ≈ Median

Choosing the Best Measure of Center Mean Commonly used & easy to understand Takes all values into account Affected by extreme values (non-resistant) Median Gives the halfway point of the data Only takes middle values into account Not affected by extreme values (resistant) Mode Gives the most frequently occurring value Only takes common values into account Not affected by extreme values (resistant) To decide which measure of center to use depends on whether or not you have an outlier as well as the shape of the data.

Choosing the Best Measure of Center Determine the best measure of center to use in each of the following situations: A shoe store trying to determine which size shoe to restock A typical house buyer trying to determine the price they should expect to pay in a particular area that has just a handful of very expensive homes. A cookie company trying to determine the amount of sales they should expect per day when looking at previous data, all of which is relatively consistent

Measures of Center from a Frequency Table

Data value (x)Frequency (f) Total40

Measures of Center from a Frequency Table Data value (x) Frequency (f) Product (fx) Total40278

Measures of Center from a Frequency Table Number of acesFrequency Total55

Measures of Center from a Frequency Table Number of aces FrequencyProduct Total55179

Measures of Center from a Frequency Table How can we make the calculator do this for us? “Stat” “Edit” Enter data values into L 1 and frequency into L 2 “Stat” “Calc” “1-Var Stats” List: L 1 (or applicable list) Frequency: L 2 (or applicable list) “Calculate”

Measures of Center for Grouped Data When data has been grouped, we will use the midpoint or mid-interval value of each class to represent all the scores within the interval. Why do we do this? We are assuming that all the scores within each class are evenly distributed throughout the interval. The mean calculated is an approximation of the true value, and we cannot do any better than this without knowing each individual data value.

Measures of Center for Grouped Data Age (years)Frequency (f)Midpoint (x) Total137

Measures of Center for Grouped Data Age (years)Frequency (f)Midpoint (x) , , , Total1375,161

Measures of Center for Grouped Data How can we make the calculator do this for us? “Stat” “Edit” Enter midpoints into L 1 and frequency into L 2 “Stat” “Calc” “1-Var Stats” List: L 1 (or applicable list) Frequency: L 2 (or applicable list) “Calculate”

Measures of Spread Describe the below graphs: Is using a measure of center enough to accurately describe a distribution?

Measures of Spread Range: Maximum value – minimum value Quartiles: Splitting the data up into 4 equal parts Interquartile range (IQR): describes how the middle 50% of the data is behaving IQR = Upper quartile – lower quartile IQR = Q 3 – Q 1 Standard deviation: measures the spread by looking at how far apart the observations are from the mean; a standard deviation with a larger value means the data is further away from the mean

Range Example. A random sample of people determined the number of cats they wished they had: 10, 12, 12, 13, 15, 16, 9, 10, 14, 11 Find the range. 16 – 9 = 7 cats

Quartiles & IQR To calculate quartiles: 1.Put the data in numerical order 2.Find the median (Q 2 ) 3.Find the median of the numbers below the median (lower quartile or Q 1 ) 4.Find the median of the numbers above the median (upper quartile or Q 3 ) To calculate the IQR (interquartile range): Upper quartile – lower quartile or Q 3 – Q 1

Quartiles & IQR Example. Calculate the median, lower quartile (Q 1 ), upper quartile (Q 3 ), and IQR: 7, 3, 1, 7, 6, 9, 3, 8, 5, 8, 6, 3, 7, 1, 9 First put the data in numerical order lower upper 1, 1, 3, 3, 3, 5, 6, 6, 7, 7, 7, 8, 8, 9, 9 Median: 6 Q 1 : 3Q 3 : 8 IQR: 8 – 3 = 5

Range, Quartiles & IQR How can we make the calculator do this for us? “Stat” “Edit” Enter data values into L 1 “Stat” “Calc” “1-Var Stats” List: L 1 (or applicable list) “Calculate” Range: maxX – minX Quartiles: Q1 Med (Q2) Q3 IQR: Q3 – Q1

Outliers To “officially” determine if a value is an outlier, we use the 1.5 X IQR criterion: Q 1 – 1.5 X IQR Q X IQR Any values beyond this range are outliers!

Measures of Spread Example. Using your GDC, calculate the median, lower quartile (Q 1 ), upper quartile (Q 3 ), and IQR of the following data: Median: 34Lower quartile: 25Upper quartile: 41 IQR: 16 Do we have an outlier? Show how you can tell. Lower outlier bound: Q 1 – 1.5 X IQR: 25 – 1.5 X 16 = 1 Upper outlier bound: Q X IQR: X 16 = is an outlier because it is beyond the upper outlier bound

Measures of Spread The problem with using the range & IQR as measures of spread is that both of them only use 2 values in the calculation. Standard deviation (s x ) takes into account the deviation of each value from the mean. It is a good measure of the dispersion of the data.

Properties of Standard Deviation (s x ) s x measures the spread about the mean and therefore should only be used when the mean is chosen as the measure of spread s x (like the mean) is not resistant s x = 0 only when there is no spread – When does this happen? When all data values are the same

Calculating Standard Deviation

Score (x) 22 – 5 = -3 (-3) 2 = 9 44 – 5 = -1 (-1) 2 = 1 55 – 5 = 0 (0) 2 = 0 55 – 5 = 0 (0) 2 = 0 66 – 5 =1 (1) 2 = 1 66 – 5 = 1 (1) 2 = 1 77 – 5 = 2 (2) 2 = 4 Total16

Standard Deviation How can we make the calculator do this for us? “Stat” “Edit” Enter data values into L 1 “Stat” “Calc” “1-Var Stats” List: L 1 (or applicable list) “Calculate” Standard deviation: σx (do NOT use Sx!)

Standard Deviation Example. 50 students were asked to total the number of points that they received on their IB diploma. Their results are shown in the table below. Using your GDC, calculate the mean and standard deviation for the boys and girls separately. Comment on what these numbers mean for each gender.

Standard Deviation Boys’ mean: 34 points Boys’ standard deviation: 1.23 points Girls’ mean: 34.3 points Girls’ standard deviation: 2.41 points Both boys & girls have a mean of about 34 points. The standard deviation for the boys is small, which implies that most boys scored close to 34 points. However, the standard deviation for girls is larger, which implies that some girls will scored much less than 34 points while others scored much more.

Standard Deviation To find the standard deviation of grouped data, use the mid-interval values. Example. Use your GDC to estimate the mean & standard deviation for this distribution of honors biology test scores. (Remember to use the midpoint values) Mean: 59.8 Standard deviation: 16.8

Choosing the Best Measures When should we use mean & standard deviation? – If the distribution is close to symmetrical When should we use median & quartiles? – If the distribution is considerably skewed