Descriptive Statistics Educational Research Chapter 12 Descriptive Statistics Gay, Mills, and Airasian 10th Edition

Topics Discussed in this Chapter Preparing data for analysis Types of descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics

Preparing Data for Analysis Issues Scoring procedures Tabulation and coding Use of computers

Scoring Procedures Instructions Types of items Standardized tests detail scoring instructions Teacher-made tests require the delineation of scoring criteria and specific procedures Types of items Selected response items - easily and objectively scored Open-ended items - difficult to score objectively with a single number as the result

Tabulation and Coding Tabulation is organizing data Coding Identifying all information relevant to the analysis Separating groups and individuals within groups Listing data in columns Coding Assigning names to variables EX1 for pretest scores SEX for gender EX2 for posttest scores

Tabulation and Coding Reliability Concerns with scoring by hand and entering data Machine scoring Advantages Reliable scoring, tabulation, and analysis Disadvantages Use of selected response items, answering on scantrons

Tabulation and Coding Coding Assigning identification numbers to subjects Assigning codes to the values of non-numerical or categorical variables Gender: 1=Female and 2=Male Subjects: 1=English, 2=Math, 3=Science, etc. Names: 001=John Adams, 002=Sally Andrews, 003=Susan Bolton, … 256=John Zeringue

Computerized Analysis Need to learn how to calculate descriptive statistics by hand Creates a conceptual base for understanding the nature of each statistic Exemplifies the relationships among statistical elements of various procedures Use of computerized software SPSS-Windows Other software packages

Descriptive Statistics Purpose – to describe or summarize data in a manner that is both understandable and short Four types Central tendency Variability Relative position Relationships

Descriptive Statistics Graphing data – a frequency polygon Vertical axis represents the frequency with which a score occurs Horizontal axis represents the scores themselves

Quiz 1 Results

Central Tendency Purpose – to represent the typical score attained by subjects Three common measures Mode Median Mean

Central Tendency Mode Median The most frequently occurring score Appropriate for nominal data Look for the most frequent number Median The score above and below which 50% of all scores lie (i.e., the mid-point) Characteristics Appropriate for ordinal scales Doesn’t take into account the value of all scores Look for the middle # (if 2 are in contention, get the mean of these 2 numbers.

Central Tendency Mean The arithmetic average of all scores Characteristics Advantageous statistical properties Affected by outlying scores Most frequently used measure of central tendency Add all of the scores together and divide by the number of Ss

Calculate for the following data points: ??= ??? Mode Median Mean

You know the central score, do you need anything else? What is the mean of the following: 10, 20, 200, 10, 20 51, 52, 53, 52, 52 Is there more we want to know about the data than just what is the middle point?

Quiz 1: Central Tendency Count: 25 Average/ Mean: 79.7 Median: 83.5

Variability Purpose – to measure the extent to which scores are spread apart Four measures Range Variance Standard deviation (there are others, but these are the only ones we are going to talk about)

Variability Range The difference between the highest and lowest score in a data set Characteristics Unstable measure of variability Rough, quick estimate Calculate What is the range of the following: 10, 20, 200, 10, 20 51, 52, 53, 52, 52

Quiz 1 Count: 25 Average: 79.7 Median: 83.5 Maximum: 93.4 Minimum: 0.0

Variability Variance The average squared deviation of all scores around the mean Characteristics Many important statistical properties Difficult to interpret due to “squared” metric Used mostly to calculate standard deviation Formula

Variance 51 - 52 = -1 52 - 52 = 0 53 - 52 = 1 10 - 52 = -42 20 - 52 = -32 200-52 = 148

Variance 51 - 52 = -12 = 1 52 - 52 = 02 = 0 53 - 52 = 12 = 1 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904

Variance 51 - 52 = -12 = 1 52 - 52 = 02 = 0 53 - 52 = 12 = 1 2 2/5=.4 Variance = .4 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 27480 27480/5 = 5496 Variance = 5496

Variability Standard deviation The square root of the variance Characteristics Many important statistical properties Relationship to properties of the normal curve Easily interpreted Formula

Standard Deviation 51 - 52 = -12 = 1 10 - 52 = -422 = 1764 52 - 52 = 02 = 0 53 - 52 = 12 = 1 2 2/5=.4; Variance = .4 __ √.4 = .63 = SD 10 - 52 = -422 = 1764 20 - 52 = -322 = 1024 200-52 = 1482 =21904 27480 27480/5= 5496= Variance ____ √5496 = 74.13 = SD

So now you know middle # and spreadoutedness How can you use that information to standardize all of the scores to have the same meaning. First set of scores has a mean of 52 and a SD of .63; second set has a mean of 52 and a SD of 74.13. How do we compare an individual score on first to an individual score on second?

Quiz 1: Variance Count: 25 Average: 79.7 Median: 83.5 Maximum: 93.4 Minimum: 0.0 Standard Deviation: 18.44

The Normal Curve A bell shaped curve reflecting the distribution of many variables of interest to educators Gives a visual way of identifying where one person’s scores fit in with the rest of the people.

Normal Curve

The Normal Curve Characteristics Fifty-percent of the scores fall above the mean and fifty-percent fall below the mean The mean, median, and mode are the same values Most participants score near the mean; the further a score is from the mean the fewer the number of participants who attained that score Specific numbers or percentages of scores fall between ±1 SD, ±2 SD, etc.

The Normal Curve Properties Proportions under the curve ±1 SD = 68%

Skewed Distributions None - even Positive – many low scores and few high scores Negative – few low scores and many high scores

Skewed Distribution Which direction are the following scores skewed: 12,4,5,13,4,4,1,3,1,3,1,3,1,5 Step 1: Reorder from lowest to highest 1,1,1,3,3,3,4,4,4,5,5,12,13 Step 2: Graph these numbers Step 3: Compare the graph to the pictures we showed above (tail goes toward the direction… tail to the right, positive; tail to the left, negative)

Skewed Distribution Example 1 3 4 1 3 4 5 1 3 4 5 12 13

Skewed Distribution Example

Measures of Relative Position Purpose – indicates where a score is in relation to all other scores in the distribution Characteristics Clear estimates of relative positions Possible to compare students’ performances across two or more different tests provided the scores are based on the same group

Measures of Relative Position Types Percentile ranks – the percentage of scores that fall at or above a given score Standard scores – a derived score based on how far a raw score is from a reference point in terms of standard deviation units z score T score Stanine

Measures of Relative Position z score The deviation of a score from the mean in standard deviation units Characteristics Mean = 0 Standard deviation = 1 Positive if the score is above the mean and negative if it is below the mean Relationship with the area under the normal curve

Measures of Relative Position T score – a transformation of a z score Characteristics Mean = 50 Standard deviation = 10 No negative scores

Measures of Relative Position Stanine – a transformation of a z score Characteristics Nine groups with 1 the lowest and 9 the highest

Measures of Relationship: Correlations Purpose – to provide an indication of the relationship between two variables Characteristics of correlation coefficients Strength or magnitude – 0 to 1 Direction – positive (+) or negative (-) Types of correlation coefficients – dependent on the scales of measurement of the variables Spearman rho – ranked data Pearson r – interval or ratio data

Measures of Relationship Interpretation – correlation does not mean causation Formula see page 316 in your text book to discuss the formula for the Pearson r correlation coefficient.

Calculating Descriptive Statistics Using SPSS Windows Means, standard deviations, and standard scores The DESCRIPTIVE procedures Correlations The CORRELATION procedure Objectives 10.1, 10.2, 10.3, & 10.4