Summation Notation, Percentiles and Measures of Central Tendency Overheads 3

Statistical Notation for Variables

Organizing Your Data Grp 1Grp 2 Obs 1 X 1 1 X 1 2 Obs 2 X 2 1 X 2 2 Obs 3 X 3 1 X 3 2 Obs 4 X 4 1 X 4 2 Obs 5 X 5 1 X 5 2 Grp 1Grp 2 Obs Obs Obs Obs Obs X 4 1 =3.00 X 5 1 =2.00 X 3 2 =10.00 X 5 2 =14.00

Sigma Notation Often, it is necessary for us to add together sets of scores, so we need a convenient way to tell someone “Add up the scores for a group of people.” In statistics, the greek symbol sigma is used to denote “add together”.

Summation Notation if there is only one group. means... “Sum the raw scores for i =1 to N”

Example Grp

Summation Notation for more than one group.

Example Grp 1Grp

Order of Operations Sum of the scores squared. Add the numbers together first, then square that sum. Sum of the squared scores. Square the number first, then add them together.

In-class Statistical Notation Problem Set (located in Course Materials)

Problem #

Problem #2 ( )( ) += 25.00

Problem # = 20.00

Problem # = 25.00

Problem # = 25.00

Problem #6 (5 2 =25) + (6 2 =36) + (4 2 =16) + …=

Problem #7 ( ) 2 = 20 2 = 400 ( ) 2 =23 2 = 529 ( ) 2 =31 2 = 961 ( ) 2 =25 2 = 625 2,515

Problem Set is different from

Shapes/Types of Distributions

How can we divide up the frequency distribution. Percentiles  A frequency distribution divided into 100 equal parts.  A percentile tells us what percent (proportion of the distribution) falls at or below the score interval of interest. Quartiles  A frequency distribution divided into four equal parts. Q1 = P25; Q2 = P50; Q3 = P75; Q4 = P99 Deciles  A frequency distribution divided into 10 equal parts. D1, D2, D3, …, D10 = P99 All of these measures are on ordinal scales.

Percentiles and the Normal Distribution X These are not equivalent halves! Note: See Handout “Location of Percentiles on a Normal Curve” in Course Materials

Percentiles and the Normal Distribution X This line must be moved to the left to form two equivalent halves! Note: See Handout “Location of Percentiles on a Normal Curve” in Course Materials

X Quartiles and the Normal Distribution Q1Q2Q3 25% P25P50 P75

Deciles and the Normal Distribution X D5 100 D D3 99 D2 98 D D5D4D3D2D

Getting a percentile rank for a particular raw score.

Getting a raw score for a specific percentile.

Measures of Central Tendency Measures of central tendency help to give information about the most likely score in a distribution. We have three ways to describe central tendency:  Mean  Median  Mode The type of measure of central tendency you should use depends on what kind of data you have.

The Mode The Mode is the score within a set of scores that appears most frequently. The Mode is appropriate for Nominal scale data. If all scores are the same then there is no Mode. If two adjacent scores both have the same, and the highest frequency, then the Mode is the average between the two scores. If two non-adjacent scores have the same and highest frequency then the group of scores is Bimodal.

The Mode… XfXf (midpoint 6.5)

The mode… X f X f f X MODE

Median The Median is the 50 th percentile in a group of scores. The Median divides the rank scores so that half of the scores fall above the median and half fall below. The Median is calculated exactly as the 50 th percentile.

The median… X f X f f X MEDIAN 50% of the distribution

Finding the median for an ungrouped frequency distribution. If there is an odd number of scores then the median is the middle score. If there is an even number of scores then the median is the halfway point between the middle most two values.

N=35 (odd number of scores) N=35/2 = 17.5 Since we do not have “half” scores, we use the 18 th scores to represent the median. Finding the median for an ungrouped frequency distribution.

The median. Xf There are a total of 35 scores, so we are looking for the interval with the 18 th score. The cumulative frequency reaches 18 in the interval of 6.00, therefore, the median is 6.00.

The median… There are a total of 16 scores, so we are looking for the that has the two middle scores (the 8 th and 9 th scores). The 8 th score is in the interval 5.00 and the 9 th score is in the interval So, the median is Xf

The Mean Mean Mean of combined groups when nj is equal for all groups Mean of combined groups when nj is not equal for all groups

Practicing Calculations: Measures of Central Tendency See Handout in Course Materials

In-class exercise: Measures of central tendency (located in Course Materials)

Properties of the mean. 1) The sum of all deviation scores around the mean will be exactly zero.

Properties of the mean. See handout: “Properties of the mean” Located in Course Materials

Properties of the mean. The sum of all deviation scores around the mean will be exactly zero. The sum of squared deviations will always be less than the sum of the squared deviations around any other point.  Least sum of squares.

The mean… X f X f f X MEAN

Location of Mean, Median, and Mode in a Distribution If a distribution is symmetrical, and unimodal, the mean, median and mode will have the same value. If a distribution is unimodal and skewed, these measures will be arranged in the order of mean, median, and mode, starting from the longest tail.  In negatively skewed distributions the mean will be less than the median.  In positively skewed distribution the mean will be greater than the median.  The difference between the mean and the median in a distribution is an indication of skewness.

The mean, median, and mode. X f X f f X Mean Median Mean Median Mode Median Mean

Central Tendency for Normal Distribution Mean15.00 Median15.00 Mode15.00

Central Tendency for Bimodal Distribution Mean15.00 Median15.00 Mode14.00 and 16.00

Central Tendency for Positively Skewed Distribution Mean13.10 Median12.00 Mode12.00

Central Tendency for Negatively Skewed Distribution Mean Median Mode18.00

SPSS- Calculating measures of central tendency Change “var” names to “group” names

SPSS- Calculating measures of central tendency

To get measures of central tendency, click “Statistics” We can run a single group (as shown) or all four groups at a time

SPSS- Calculating measures of central tendency To find the raw score that corresponds to the 65 th percentile, (1) check box, (2) type in percentile, (3) click “add”

SPSS- Calculating measures of central tendency See Handout for “Output for Central Tendency” in Course Documents

SPSS- Calculating measures of central tendency

If there is more than 1 mode, SPSS reports the lowest one and tells you other modes exist

SPSS- Calculating measures of central tendency An alternate way of obtaining the measures of central tendency is with “Descriptives”

SPSS- Calculating measures of central tendency

The Descriptives table puts the group variables in rows and statistics in columns The means The means of each group The minimums and maximums are the lowest and highest scores in each group

SPSS- Calculating measures of central tendency A third option for obtaining the measures of central tendency is with “Explore”

SPSS- Calculating measures of central tendency Transfer all four group variables to the dependent list and click “ok”

SPSS- Calculating measures of central tendency Explore provides the mean and the median, not the mode