Sec 60-61 Descriptive & Inferential Statistics: Their Value in Research

Question 1

Descriptive Statistics – Value in Research

Answer 1

DESCRIPTIVE STATISTICS – measures of central tendency and measures of variability: (ex: Mean, Median, Range, Standard Deviation, Percent, Pearson r, Coefficient of Determination (r²))

• Help researchers obtain OVERVIEWS of selected characteristics of the groups they study.
• Make it relatively easy to COMPARE two or more groups on the characteristics (i.e., variables) being studied.
• Allow researchers to COMMUNICATE their findings quickly and efficiently.

Standard 5-Step Process to Describe Data:

1. DETERMINE THE SCALE of MEASUREMENT – Remember the types of Scales:
   
   • NOMINAL – Naming Level (Name political parties, tree types, and regions of the USA)
   • ORDINAL – puts date IN ORDER from high to low, but it does not specifically indicate how much higher or lower one value is in relation to another.
     
     • Note: that if scores only RANK students (and do NOT measure them with equal units (eg. raw score on a test), the scale would be ordinal
   • INTERVAL – indicates how much the values differ from each other. It is helpful to think of these as the EQUAL DISTANCE levels and NO ABSOLUTE ZERO (Ex: IQ has no absolute zero as a value)
     
     • Note: multiple-choice test scores use an interval scale of measurement. Assumes each correct answer on the test represents the same amount of knowledge as each other correct answer.
     • Also, note: that the assumption that the scores are at equal intervals will influence the selection of additional statistics in subsequent steps.
   • RATIO – indicates how much the values differ from each other. It is helpful to think of these as the equal distance levels but it DOES HAVE an ABSOLUTE ZERO (Ex: We know where the zero is on a tape measure when measuring distance.)
     
     • 2. EXAMINE THE SHAPE of the DISTRIBUTION using one of the following:
   • A statistical table such as a frequency distribution
   • Histogram
   • Frequency Polygon
2. _MEAN or MEDIAN (or Pearson *r* and the Coefficient of Determination *r²*)_ –
   
   • If the distribution is at least roughly SYMMETRICAL and not highly skewed, calculate the MEAN.
   • If the scale of measurement is ORDINAL, such as RANK or if the distribution is highly skewed, select the MEDIAN
   • Compute the average (i.e., mean or median) for all 325 students. Also, compute the average for each group the administrators want to compare, such as the students at each of the three schools (Washington, Franklin, and Bishop), so that the averages can be compared across schools.
   • If you’re comparing two sets of data for a relationship, compute PEARSON r and the COEFFICIENT of DETERMINATION r².
3. PERCENTILE RANK and STANDARD SCORES – If the administrators want to use the test scores to determine how each student performed in relation to the other students in the school or the district, compute PERCENTILE RANK and/or STANDARD SCORES for each student.
4. EXAMINE THE VARIABILITY in the SCORES:
   
   • If the MEDIAN was selected as the average, compute the RANGE and INTERQUARTILE RANGE.
   • If the MEAN was selected as the average, compute the STANDARD DEVIATION.
   • Also, Calculate MEAN/STANDARD DEVIATION - MEDIAN/RANGE for each SUBGROUP so that you can compare them to each other as well as to the entire group.

By following the five steps listed in this section, the school administrators will be able to:

1. Obtain an OVERVIEW of the students’ abilities to add one-digit numbers-at either the district, school, or classroom level.
2. COMPARE various groups of students within the school district, and…
3. …concisely COMMUNICATE important information, such as individual students’ percentile ranks as well as information about the groups’ averages and variability.

Question 2

Inferential Statistics – Value in Research

Answer 2

INFERENTIAL STATISTICS are used to INTERPRET descriptive statistics in light of sampling errors. (Ex: Confidence Intervals, Tests for Statistical Significance)

1. The first major value of inferential statistics: They help us to INTERPRET descriptive statistics that estimate population values (such as the mean of a population) based on random samples.
   
   • So Descriptive Statistics DESCRIBE a set of data using averages (mean, median, mode) and variability (range, interquartile range, standard deviation).
   • Inferential Statistics come into play when you are using a sample. Inferential statistics, such as confidence intervals, are calculated using the Descriptive Statistics for that sample in order to INTERPRET how that sample might reflect the actual population.
   • Ex: The MLB wanted to test the baseball knowledge of all high school students in the USA. They took a simple random sample of these students, and miraculously, all 1,000 of those selected agreed to take a 50-item baseball-knowledge multiple-choice test. The mean score for the sample was 35.0. but because it was only a sample and not the population, there was a decent probability that this sample mean did not reflect the population mean.
     
     • To give the sample mean more meaning, the researchers also reported that a safer estimate of the high schoolers’ baseball knowledge was the 95% confidence interval for the mean, (31.0 to 39.0). Thus, the MLB was able to have 95% confidence that the true mean (which would have been obtained if the researchers had tested all high schoolers in the USA) would have been between 31.0 and 39.0.
     • Based on this information, the MLB hired consultants to advise the board on how to increase students’ knowledge of baseball.
2. The second major value of inferential statistics: Inferential statistics help in the INTERPRETATION of DIFFERENCES in descriptive statistics AMONG VARIOUS SAMPLES drawn at random. (Ex: Tests of Statistical Significance)
   
   • So, given 2 different samples, you can test to see if the DIFFERENCE between the result (Ex: means) from the two samples are statistically significant (That you can REJECT THE NULL HYPOTHESIS) and conclude that the difference between the results is NOT simply the result of simple random sampling error. (i.e. the difference was greater than what would be expected on the basis of random error alone. Hence, the result is significant.)
     
     • Ex: Two programs were set up to help kids develop their unicycle skills. A random sampling of 12-year-olds was chosen for each program. When all was complete, those who were in Program A were able to ride the unicycle an average of 65 seconds without falling off. Those who were in Program B were able to stay on for just 62 seconds on average.
       
       • By conducting an inferential test of statistical significance, Program coordinators determined the probability that the difference between 65 seconds and 62 seconds was not statistically significant.
       • In light of this result, the directors concluded that the two programs appeared to be about equal in their effectiveness. Given this information, they selected Program B for future use because it was far less expensive than Program A.

• In summary, inferential statistics help in the interpretation of descriptive statistics in light of possible errors created by random sampling.
  
  • Since studying only random samples is usually more efficient than studying whole populations, and inferential statistics help researchers make inferences about what populations are like based on random samples drawn from those populations.