Measures of Dispersion

Transcription

1 Measures of Dispersion 6-3 I Will... Find measures of dispersion of sets of data. Find standard deviation and analyze normal distribution.

2 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

3 To Find the Range: Steps: 1. Find the highest and lowest values 2. Find the difference between the highest and lowest values Max = 85 Min = 23 Range = = 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

4 To Find the Mid-Range of a Data Set: Steps: Example: 1. Find the highest and Lowest values. 2. Find the mean of the highest and lowest values. 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.

5 To Make a Relative Frequency Distribution: Steps: Example: 1. Make a frequency distribution 2. 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. Check: = 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.

6 To Make a Cumulative Relative Frequency Distribution Steps: 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. Quartiles Boundaries 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.

7 Example: Quartiles Median of all Data ( ) 2 = 80 Median of Lower Half 70 Median of Upper Half 90 Q1 = Below 70 Q2 = 70 to 80 Q3 = 80 to 90 Q4 = Above 90 LOWER HALF UPPER HALF 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!

8 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

9 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 # = Data Set 2,2,4,4,4,5,5,6,6,8,8,8,9,12,12,12,14,15,20,20 Total Items: 20 Find the following percentiles: (a) 25% (b) 80% 6th item is 5. 16th item is Find the data item that corresponds to the ordinal number found in step 3. Day 2: Standard Deviation & Normal Distribution

10 Deviation from the Mean the difference between the value and the mean gives us a clearer picture of the data set than range does Steps: 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 Values Values Smaller than the Mean negative below or less than mean Absolute Value of the Sum of the Deviations BELOW the Mean = Absolute Value of the Sum of the Deviations ABOVE the Mean Values Larger than the Mean positive above or greater than mean so The sum of ALL deviations from the mean is ZERO.

11 Standard Deviation And Variance Deviation- 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:

12 Example: Standard Deviation Example: Standard Deviation Mean: 394 mm Values x 1 Deviations from mean Squares of the deviations from the mean ( ) = 206 (206) 2 = 42, = 76 (76) 2 = 5, = -224 (-224) 2 = 50, = 36 (36) 2 = 1, = -94 (-94) 2 = 8,836 Sum: 1,970 Sum: 0 Sum: 108,520

13 Example: Standard Deviation Data Distribution

14 Normal Distribution Normal Distribution

15 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.

16 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 = ( ) 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! s = = 0.15m ( ) 0.15 = 3 Standard Deviations 1 2 3