Statistics Measures

Chapter 15

Sections 15.5-15.8

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Mean Measurement15.5

• To find the mean measurement gather several measurements.

• The more measurements, the more accurate the mean is

normally.

• The mean measurement is then found by dividing the sum of

these measurements by the number of measurements taken.

Mean Measurement

• A machinist measured the thickness of a metal disk with a micrometer at four different places. She found the following values:

• 2.147 in., 2.143 in., 2.151 in., 2.148 in.

• Find the mean measurement.

Example 1

• Step 1: Add the measurements:Example 1

cont’d

• Step 2: Divide the sum of the measurements by the number of measurements.

• So the mean measurement is 2.147 in.

• Note that the mean measurement is written so that it is rounded to same place as each of the measurements.

Example 1cont’d

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Other Average Measurementsand Percentiles15.6

• Since mean measurements are greatly affected or made less

accurate by numbers that are much higher or lower than the

rest of the group, there are other procedures to determine an

average measurement besides finding the mean measurement.

• The median measurement is the measurement that falls in the

middle of a group of measurements arranged in order of size.

• That is, one-half of the measurements are larger than or equal

to the median, and one-half of the measurements are less than

or equal to the median.

Other Average Measurements and Percentiles

• When there is an even number of

measurements, there is no one middle

measurement.

• In this case, the median is found by taking the

mean of the two middle measurements. (Add

the two middles and divide by 2.)

Other Average Measurements and Percentiles

• Find the median measurement of the following set of measurements. 54, 57, 59, 55, 53, 57, 50, 56

• Step 1: Arrange the measurements in order of size.

59 57 (Note: since there are two measures of 57

57 both need to be listed!)

56 55 54 53 50

Example 2

• Step 2: Since there are eight measurements,

find the mean of the two middle

measurements.

• So the median measurement is 55.5.

Example 2cont’d

• Another kind of measure used is the mode. The mode is the

measurement that appears most often in a set of

measurements.

• In Example 2, 57 is the mode, because two of the

measurements have this value. However, the mode can present

problems.

• Some mathematicians believe there can be more than one

mode if more than one number appears most often.

• The mode may or may not be near the middle.

Other Average Measurements and Percentiles

• Related to averages is the idea of measuring the position of a

piece of data relative to the rest of the data.

• Percentiles are numbers that divide a given data set into 100

equal parts.

• The nth percentile is the number Pn such that n percent of the

data (ranked from smallest to largest) is at or below Pn.

Other Average Measurements and Percentiles

• For example, if you score in the 64th

percentile on some standardized test, this

means that you scored higher than 64% of

those who took the test and you scored lower

than 36% of those who took the test. In other

words, your score sits at the 64th spot in the

order of test scores.

Other Average Measurements and Percentiles

• The following list gives 50 pieces of ranked data (ranked from smallest to largest).

• a. Find the 98th percentile.

• b. Find the 75th percentile.

• c. Find the 26th percentile.

Example 4

• a. The 98th percentile is 172

98/100 = ?/50 cross multiply and divide to find that 98th percentile is the 49th piece of data. Look at the chart to find the 49th piece is 172.

• b. The 75th percentile is 142 (the 38th piece of data)

Actually when we find 75/100 = ?/50 we get 37.5 but since a piece of data in the chart can not be divided in half we round up to 38 and look for the 38th piece which is 142.

• c. The 26th percentile is 51 (the 13th piece of data)

• Can you explain why?

Example 4 – Solution

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Range and Standard Deviation15.7

• The mean measurement gives the technician the average value of a group of measurements, but the mean does not give any information about how the actual data vary in values.

• Some type of measurement that gives the amount of variation is often helpful in analyzing a set of data.

• One way of describing the variation in the data is to find the range.

• The range is the difference between the highest value and the lowest value in a set of data.

Range and Standard Deviation

• Find the range of the following measurements:

• 54°, 57°, 59°, 55°, 53°, 57°, 50°, 56°

• The range is the difference between the highest value, 59°, and the lowest value, 50°.

• The range is 59° – 50° = 9°.

Example 1

• The range gives us an idea of how much the data are spread

out, but another measure, the standard deviation, is often more

helpful.

• The standard deviation tells how the data typically vary from

the mean. Suppose we are given data sets

• A = 4, 5, 5, 6 and B = 2, 3, 7, 8.

• Both sets have a mean of 5, but the data in set A are “nearer”

to the mean than are the data in set B.

Range and Standard Deviation

• A mathematical way of describing this is to use standard deviation.

• There are two types of standard deviation: population standard deviation, denoted by , and sample standard deviation, denoted by s.

• Sample standard deviation is calculated using each piece of data and is useful for small groups of data.

• Population standard deviation groups data into sets and approximates the mean and the standard deviation from them. It is useful for larger groups of data.

• The formula for finding sample standard deviation is (text pg. 535)

Range and Standard Deviation

• In set A,

• In set B,

• This means that for set A “normal” values can be anywhere from 5.82

down to 4.18 where in set B “normal” values can be anywhere from 7.9

down to 2.1 so set B has more variation in the values and that is OK.

Range and Standard Deviation

• Find the sample standard deviation for the following data:

• 54°, 57°, 59°, 55°, 53°, 57°, 50°, 56°

• Step 1: Find the mean.

Example 2

•Step 2: Find the difference between each piece of data and the mean.

54 – 55.1 = –1.1

57 – 55.1 = 1.9

59 – 55.1 = 3.9

55 – 55.1 = –0.1

53 – 55.1 = –2.1

57 – 55.1 = 1.9

50 – 55.1 = –5.1

56 – 55.1 = 0.9

Example 2

• Step 3: Square each difference and find the sum of the squared amounts.

• (–1.1)2 = 1.21

• (1.9)2 = 3.61

• (3.9)2 = 15.21

• (–0.1)2 = 0.01

• (–2.1)2 = 4.41

• (1.9)2 = 3.61

• (–5.1)2 = 26.01

• (0.9)2 = 0.81

• 54.88

Example 2cont’d

• Step 4:

• Using the formula

Example 2cont’d

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Grouped Data15.8

• Finding the mean of a large number of measurements can take much time and can be subject to mistakes. Grouping the measurements (the data) can make the work in finding the mean much easier.

• Grouped data are data arranged in groups that are determined by setting up intervals. An interval contains all data between two given numbers a and b.

• We will show such an interval here by the form a–b. For example, 2–8 means all numbers between 2 and 8.

Grouped Data

•The number a is called the lower limit and b is called the upper limit of the interval.

•The number midway between a and b, , is called the midpoint of the interval. In the above example, the lower limit is 2, the upper limit is 8, and the midpoint is

•While there are no given rules for choosing these intervals, the following general rules are helpful.

•General Rules for Choosing Intervals for Grouped Data

•1. The number of intervals chosen should be between 6 and 20.

Grouped Data

•2. The length of all intervals should be the same and should always be an odd number.

•3. The midpoint of each interval should have the same number of digits as each of the measurements that fall within that interval. The lower limit and the upper limit of each interval will have one more digit than the measurements within the interval.

• In this way, no actual measurement will have exactly the same value as any of these limits. It will therefore be clear to which interval each measurement belongs.

Grouped Data

•4. The lower limit of the first interval should be lower than the lowest measurement value, and the upper limit of the last interval should be higher than the highest measurement value.

•Once the intervals have been chosen, form a frequency distribution. A frequency distribution is a list of each interval, its midpoint, and the number of measurements (frequency) that lie in that interval.

Grouped Data

•Make a frequency distribution for the recorded high temperatures for the days from November 1 to January 31 as given in Table 15.1.

Example 1

Example 1 cont’d

Example 1• First, choose the number and size of the group intervals to be

used. We must have enough group intervals to cover the range of the data (the difference between the highest and the lowest values).

• Here, the range is 55 – 2 = 53.

• Since 53 is close to 54, let us choose the odd number 9 as the interval length.

• This means that we will need 54 9 = 6 group intervals. This satisfies our general rule for the number of intervals.

cont’d

Example 1• Our first interval is 1.5–10.5 with a midpoint of 6.

• Here, 1.5 is the lower limit and 10.5 is the upper limit of the interval.

• We then make the frequency distribution as shown in Table 15.2.

cont’d

•To find the mean from the frequency distribution, (a) multiply the frequency of each interval by the midpoint of that interval, xf ;(b) add the products of xf ; and (c) divide by the number of data, sum of f.

Grouped Data

•Find the mean of the data given in Example 1.

•A frequency distribution table (Table 15.3) gives the information for finding the mean.

Example 2

• The mean temperature is found as follows.Example 2

cont’d