Measures of Dispersion

6-3

I Will...● Find measures of dispersion of sets of data.● Find standard deviation and analyze normal distribution.

Day 1: Dispersion

Vocabulary● Measures of Variation (Dispersion or Spread)- used to describe the

variability in a sample or population.○ Example: Range

○ Important because it gives us an idea how well the measures of central tendency ( ex. Mean or Median) represent the data.

○ Big Spread = Mean/Median NOT very representative of the data○ Small Spread = Mean/Median IS very representative of the data

● Range- The difference between the highest value and the lowest value in a set of data.

○ Does not tell us whether the highest or lowest values are typical values or outliers

To Find the Range:Steps:

1. Find the highest and lowest values2. Find the difference between the highest

and lowest values

Max = 85 Min = 23

Range = 85 - 23 = 62

Vocabulary● Mid-Range- the mean of the lowest and highest values of the data set

and is a measure of central tendency○ Highly sensitive to outliers○ ONLY useful when the data set has NO outliers

● Outlier- a data item that is unusually large or unusually small when compared to the other data items

To Find the Mid-Range of a Data Set:Steps:

1. Find the highest and Lowest values.

2. Find the mean of the highest and lowest values.

Example:

Vocabulary● Relative Frequency- the fraction of times that a particular category or

class in a frequency distribution occurs.○ The SUM of the relative frequencies for a distribution is 1.

● Cumulative Frequency Distribution- made by adding the frequency of a given class to the accumulated frequency of the previous classes.

○ The cumulative frequency of the LAST class is the TOTAL number in the data set.

● Cumulative Relative Frequency- the accumulation of the relative frequencies.

○ The cumulative relative frequency of the LAST class in a distribution is 1.

To Make a Relative Frequency Distribution:Steps:

1. Make a frequency distribution2. Find the relative frequency of each class

or category by dividing the frequency by the total number in the data set.

● Express as a fraction in lowest terms or a decimal equivalent.

3. To check the distribution, the sum of all the relative frequencies is 1.

Example:

Check:0.49 + 0.37 + 0.12 + 0.02 = 1 ✔

To Make a Cumulative Frequency Distribution:Steps:

1. Make a frequency distribution.

2. The cumulative frequency of the first (lowest) class is the same as the frequency of the class.

3. The cumulative frequency of each subsequent class is the frequency of the class plus the sum of the frequencies of all previous classes.

4. The cumulative frequency of the last (highest) class is equal to the total items in the data set.

To Make a Cumulative Relative Frequency DistributionSteps:

1. Make a relative frequency distribution.

2. The cumulative relative frequency of the first (lowest) class is the same as the relative frequency.

3. The cumulative relative frequency of each subsequent class is the relative frequency of the class plus the sum of the relative frequencies of all previous classes.

4. The cumulative relative frequency of the last (highest) class is 1.

QuartilesBoundaries that divide the data set into four parts.

● Three boundaries will divide the data into four parts.

● Are a useful measure of spread because they are much less affected by outliers or a skewed data set

*there is more than one method for finding these boundaries (we will go over one method)

To Divide a Set of Data into Quartiles:

1. Find the median of the data set.a. This divides the data into two halves (lower

half and upper half)

2. Find the median of each half of the data.

3. The median of the lower half of data is the boundary for the LOWER Quartile.

4. The median of the upper half of data is the boundary for the UPPER Quartile.

Example: QuartilesMedian of all Data(80 + 80) ÷ 2 = 80

Median of Lower Half70

Median of Upper Half90

Q1 = Below 70

Q2 = 70 to 80

Q3 = 80 to 90

Q4 = Above 90

LOWERHALF

UPPERHALF

Vocabulary● Ordinal Numbers- a number that identifies an ordered position

Example:. first, second, third

● Inner Quartiles- the second and third quartiles (the quartiles in the middle)

● Inner-Quartile Range- the difference between the upper-quartile boundary and the lower quartile boundary (Q3 -Q1= IQR)

○ Half the data will be within this range

○ Useful in examining data because we eliminate all the outliers!

Inner-Quartile Range

Steps:

1. Arrange data in ascending order.

2. Find the boundary for the lower quartile.

3. Find the boundary for the lower quartile.

4. Find the difference between the upper quartile boundary and the lower quartile boundary.

Vocabulary● Percentiles- boundaries that

determine a data item (score) below which a certain percent of the data will fall.

○ Related to Quartiles■ Q1 = 25th percentile■ Q2 = 50th percentile■ Q3 = 75th percentile■ Q4 = 100th percentile

● Percentile Rank- the percent of data items that is less than or equal to the given data item

To Find a Specified Percentile:Steps:

1. Arrange the data in ascending order.

2. Assign each data item an ordinal number that corresponds to each data point.

○ Lowest data point = 1st ordinal #

3. Find the ordinal number (n) that corresponds to the given percentile rank by using the formula.

corres. ordinal # =

4. Find the data item that corresponds to

the ordinal number found in step 3.

Data Set2,2,4,4,4,5,5,6,6,8,8,8,9,12,12,12,14,15,20,20

Total Items: 20

Find the following percentiles:

(a) 25% (b) 80%

6th item is 5. 16th item is 12.

Day 2: Standard Deviation & Normal Distribution

Deviation from the Mean● the difference between the value and

the mean● gives us a clearer picture of the data set

than range doesSteps:

1. Find the mean of the data set.

2. Find the amount that each data value deviates or is different from the mean.

deviation from the mean = data value - mean

Understanding the ValuesValues Smaller than the Mean

● negative● below or less than mean

Values Larger than the Mean

● positive● above or greater than mean

Absolute Value of the Sum of the Deviations BELOW the Mean = Absolute Value of the Sum of the Deviations ABOVE the Mean

so

The sum of ALL deviations from the mean is ZERO.

Standard Deviation And VarianceDeviation- HOW FAR FROM THE NORMAL

Variance- The average of the squared differences from the mean

-Calculate by: taking the sum of the squared deviations divided by one less than the number of values.

Standard Deviation- A measure of how spread out numbers are.

-Bigger standard deviation = numbers more spread out (not typical data set)

-Smaller standard deviation = numbers closer together (typical data set)

-Calculate by: taking the square root of the variance.

To Find the Standard Deviation:1. Find the mean,

2. Find the deviation of each value from the mean:

3. Square each deviation: ( )2

4. Find the sum of the squared deviations ∑( )2

5. Divide the sum of the squared deviations by one less than the number of values in the data set. The quotient is called the sample variance:

6. Find the sample standard deviation by taking the square root of the sample variance:

Example: Standard Deviation

Example: Standard DeviationValues x1

Deviations from mean Squares of the deviations from the mean ( )2

600 600 - 394 = 206 (206)2 = 42,436

470 470 - 394 = 76 (76)2 = 5,776

170 170 - 394 = -224 (-224)2 = 50,176

430 430 - 394 = 36 (36)2 = 1,296

300 300 - 394 = -94 (-94)2 = 8,836

Sum: 1,970 Sum: 0 Sum: 108,520

Mean: 394 mm

Example: Standard Deviation

Data Distribution

Normal Distribution

Normal Distribution

When We Calculate the Standard Deviation We Find That (Generally)

To Solve Applied Problems:1. Locate the mean and the desired values on the normal curve.

# standard deviations from (above or below) the mean

2. Highlight desired regions of the normal curve based on the conditions of the problem.

3. Add percents associated with the highlighted regions in step 2.

Example: 95% of students at school are between 1.1m and 1.7m tall.

Assuming this data is normally distributed can you calculate the mean and standard deviation?

Mean = (1.7 + 1.1) ÷ 2 = 1.4m

95% represents 2 standard deviations.

Example: In that same school one of your friends is 1.85m tall

Using your graph….1.85 m is 3 standard deviations away.

You can also calculate it!

1 2 3

(1.85 - 1.4) ÷ 0.15 =3 Standard Deviations

s = 1.55- 1.4 = 0.15m