![33Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Confidence Interval (Cont’d)
To calculate a 95% confidence interval
around a mean, for example:
Calculate the mean
Calculate the standard error of the mean
Calculate the degrees of freedom (df) [df = n – 1]
Look up the two-tailed t-value for p < 0.05](https://image.slidesharecdn.com/chapter022-150722191455-lva1-app6892/85/Chapter-022-33-320.jpg)
Chapter 022
- 1. 1Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Chapter 22 Using Statistics To Describe Variables
- 2. 2Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Using Statistics to Describe Variables Two major classes of statistics Descriptive statistics • To reveal characteristics of the sample dataset Inferential statistics • To gain information about effects in the population being studied
- 3. 3Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Using Statistics to Describe All quantitative research uses descriptive statistics For description of the sample For initial description of variables For analysis of the primary research problem Descriptive statistics for descriptive research Inferential statistics for interventional and correlational research
- 4. 4Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Using Statistics to Summarize Data Terms: the number of elements in a sample is the “n” of the sample Data set: 45, 26, 59, 51, 42, 28, 26, 32, 31, 55, 43, 47, 67, 39, 52, 48, 36, 42, 61, 57 n = 20 Descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion
- 5. 5Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Frequency Distributions Table or figure (line graph, pie chart, etc.) Continuous variable: the higher numbers represent more of that variable, and the lower numbers represent less of that variable
- 6. 6Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Frequency Table Listing every possible value in the first column of numbers, and the frequency (tally) of each value as the second column of numbers Data set: 45, 26, 59, 51, 42, 28, 26, 32, 31, 55, 43, 47, 67, 39, 52, 48, 36, 42, 61, 57 (ages) Sort from lowest to highest values Tally each value
- 7. 7Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Ungrouped Frequency Distribution List all categories of the variable on which they have data, and tally each datum on the listing Age Frequency 26 2 28 1 31 1 32 1 36 1 39 1 42 2 43 1 45 1 47 1 48 1 51 1 52 1 55 1 57 1 59 1 61 1 67 1
- 8. 8Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Grouped Frequency Distribution Categories are grouped into ranges Ranges must be mutually exhaustive and mutually exclusive Age Frequency 20 - 29 3 30 - 39 4 40 - 49 6 50 - 59 5 60 - 69 2
- 9. 9Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Grouped Frequency Distribution with Percentages Adult Age Range Frequency (f) Percentage (%) Cumulative Percentage 20 – 29 3 15 15 30 – 39 4 20 35 40 – 49 6 30 65 50 – 59 5 25 90 60 – 69 2 10 100 Total 20 100
- 10. 10Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Frequency Distributions Presented in Figures Graphs Charts Histograms Frequency polygons
- 11. 11Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Line Graph
- 12. 12Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Frequency Table of Smoking Status Smoking Status Frequency Percent Current smoker 1 10 Former Smoker 6 60 Never Smoked 3 30 Total 10 100
- 13. 13Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Histogram of Smoking Status
- 14. 14Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Measures of Central Tendency Statistics that provides the center or hallmark value of a data set Mode Median (MD) Mean
- 15. 15Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Mode The most common value in a data set Bimodal: two modes exist Multimodal: more than two modes
- 16. 16Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Median (MD) The middle value in the data set (after sorting values from lowest to highest) If the “n” is even, the two values in the middle are averaged The 50th percentile
- 17. 17Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Mean Arithmetic average of all a variable's values Most commonly reported measure of central tendency Sum of the scores divided by the number of values in the data set Formula:
- 18. 18Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. When to Use Mean Mean: normally distributed values measured at the interval or ratio level Ordinal level data from a rating scale If The n is large The data are normally distributed Small values denote very little of the measured quantity; large ones denote a lot Mean is sensitive to extreme scores such as outliers
- 19. 19Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. When to Use Median and Mode Median: used for non-normal distributions with small n Mode: used for nominal values
- 20. 20Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Using Statistics to Explore Deviations in the Data Using measures of central tendency to describe the nature of a data set obscures the impact of extreme values or deviations in the data Measures of dispersion, provide important insight into nature of the data
- 21. 21Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Measures of Dispersion Quantifications of how tightly clustered around the mean the sample is: Tightly clustered = fairly homogeneous Widely dispersed = heterogeneous Range Difference score Variance Standard deviation
- 22. 22Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Range Presented in two ways: The lowest score and the highest score (2 through 17) The difference between the highest and the lowest score (range of 15)
- 23. 23Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Difference Score Subtract the mean from each score Sometimes referred to as a deviation score The difference score is positive when score is above the mean, and negative when score is below the mean The total of all the difference scores is zero Formula:
- 24. 24Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Mean Deviation Average difference score, using the absolute values Example:
- 25. 25Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Variance (s²) Variance commonly used “s2” is used to represent a sample variance “σ2” is used to represent population variance Always a positive value, has no upper limit Bigger variances = more spread Formula:
- 26. 26Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Standard Deviation (s) Square root of the variance Sometimes reported as SD Most commonly reported measure of dispersion Formula:
- 27. 27Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. The Normal Curve
- 28. 28Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. The Normal Curve (Cont’d) Represents the frequency distribution of a variable that is perfectly normally distributed Signifies: The mean is the most commonly occurring value There are just as many values above the mean as there are below the mean When frequency table is constructed, values are perfectly symmetric 68% of values are –1 to +1 standard deviations from mean 95% of values are –2 to +2 standard deviations from mean
- 29. 29Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. z-Score
- 30. 30Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. z-Score (Cont’d) Synonymous with a standard deviation unit A z value of 1.0 represents 1 standard deviation unit above the mean A z value of –1.0 represents 1 standard deviation unit below the mean Formula:
- 31. 31Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Sampling Error Described by the statistic “standard error” Standard error of the mean is calculated to determine the magnitude of the variability associated with the mean Formula: where = standard error of the mean s = standard deviation n = sample size
- 32. 32Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Confidence Interval Determines how closely a sample value approximates a population value Can be created for many statistics, such as a mean, proportion, and odds ratio Using a table of statistical values, the t-value is accessed, for the desired interval, usually 95%
- 33. 33Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Confidence Interval (Cont’d) To calculate a 95% confidence interval around a mean, for example: Calculate the mean Calculate the standard error of the mean Calculate the degrees of freedom (df) [df = n – 1] Look up the two-tailed t-value for p < 0.05
- 34. 34Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc. Degrees of Freedom The number of independent pieces of information that are free to vary For confidence interval, the degrees of freedom (df) are n – 1 This means that there are n – 1 independent observations in the sample that are free to vary (to be any value) to estimate the lower and upper limits of the confidence interval