Variance & standard deviation Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics

You should be able to… Find and analyse measures of spread (variance and standard deviation) for discrete data, and grouped discrete or continuous data. Obtain the standard deviation, and indirectly the variance, from a GDC.

Low & High Standard Deviation Low standard deviation shows that the data points tend to be very close to the mean. High standard deviation indicates that the data is spread out over a large range of values.

Variance and standard deviation Variance is calculated by finding the square of the distance each piece of data is from the mean. The standard deviation is the square root of the variance. These measures are an important indication to the spread of data. Unlike the inter-quartile range the standard deviation takes into account every piece of data. Variance Standard deviation

Comparing data Two basketball players record the number of points scored in their last 7 matches. Calculate standard deviations for both players. Player A Player B x 12 16 10 20 22 17 15 x 7 9 12 31 22 9 81 -4 16 7 49 -6 36 4 16 4 16 15 225 6 36 6 36 1 1 6 36 -1 1 7 49 What does this tell us about the players’ consistency?

Using the GDC Player A Player B Finding standard deviation with a GDC: x 12 16 10 20 22 17 15 x 7 9 12 31 22 Use a GDC to find the standard deviation for player B. In the STAT mode enter the values into a list, in this case list1 CIS U16 Boys Basketball, 2012

Grouped data without a GDC When using grouped data, the formula for finding standard deviation the formula is, Score Frequency 1-10 3 11-20 9 21-30 16 31-40 23 41-50 12 51-60 7 Centre 5.5 16.5 2268.75 15.5 139.5 2756.25 25.5 408 900 35.5 816.5 143.75 45.5 546 1875 55.5 388.5 3543.75 Verify your answers by using a GDC. Remember to make List 1 the mid-points and List 2 the frequency.

Question The table represents the weight, W, in grams, of 80 packets of roasted peanuts. Find an estimate of the standard deviation of the weight.

Question continued… Since the data is organized into class intervals, we use the mid-interval values as representative scores. You are also expected to use your GDC to find this estimate. Using the LIST feature, we have:

Question continued… With the single variable option on your GDC we can find all the relevant statistics. The value of the standard deviation is 7.41 correct to three significant figures.

Did you know? The standard deviation is a way of measuring how exceptional an individual in a population is. For example, IQ scores among adults are shown in this distribution. This kind of bell-shaped distribution is called “normal”. The yellow bars show one, two, and three standard deviations from the mean. Did you know that MENSA considers people whose IQs are 2 standard deviations above the mean?

Be prepared… Clearly identify the mid-interval values when calculating estimates for the mean and the standard deviation of grouped data.

Answers: (a) 2. 47; (b) the mean has had 100 added to it; (c) 2 Answers: (a) 2.47; (b) the mean has had 100 added to it; (c) 2.47; (d) the standard deviation remains the same. This is because the standard deviation only measures the spread of the numbers, and that remains constant if the same number is added to each item in the list; (e) the mean is doubled; (f) 4.94; and (g) the variance will be multiplied by 4 because the variance is the standard deviation squared.

Important Note You should use a GDC to calculate the population standard deviation and variance.

Important Note You may be expected to use these rules in your exam.

Properties of Standard Deviation Standard deviation is only used to measure __________________ of a data set. Can standard deviation be negative? Is standard deviation sensitive to outliers? Can a single outlier change the standard deviation? For data with approximately the same mean, the _______ the spread, the _______ the standard deviation. If all values of a data set are the same, then the standard deviation is ______.

Properties of Standard Deviation Standard deviation is only used to measure spread around the mean of a data set. Standard deviation is never negative. Standard deviation is sensitive to outliers—a single outlier can increase the standard deviation and distort the representation of spread. For data with approximately the same mean, the greater the spread, the greater the standard deviation. If all values of a data set are the same, then the standard deviation is zero.

You should know… The numerical value of the standard deviation gives an indication of how widely the data are spread about the mean. An estimate of the standard deviation of grouped data can be found by using the mid-interval value as a representative score.