Programme for the International Assessment of Adult Competencies Series
Skills in Canada: First Results from the Programme for the International Assessment of Adult Competencies (PIAAC)
Notes to Readers
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Means and confidence intervals
Throughout this report charts and tables are employed to convey study results to a broad non-technical audience and to provide a source of informative displays that readers may use for their own purposes. To satisfy the more technical reader data table for all charts is provided in the statistical annex to this report.
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Given the comparative nature of the PIAAC study, those responsible for the design of the study and its implementation went to great lengths to establish the validity, reliability, comparability and interpretability of estimates, and to control and quantify errors that might interfere with or bias interpretation. Notes to figures and tables are used to alert readers whenever errors have been detected that might affect interpretation.
The data values presented in this report are estimated from representative but complex samples of adults from each country. Consequently there is a degree of sampling error that must be taken into account. Additionally, there is a degree of error associated with the measurement of skills because they are estimated on the basis of responses to samples of test items. Thus a statistic, called the standard error, is used to express the degree of uncertainty associated with both sampling and measurement error.
When comparing scores among countries, provinces, territories or population subgroups, the degree of error in each average score should be considered in order to determine if the averages really are different from each other. Standard errors and confidence intervals may be used as the basis for performing these comparative statistical tests. Such tests can identify, with a known probability, whether there are actual differences in the populations being compared. For example, when an observed difference is significant at the 0.05 level, it implies that the probability is less than 0.05 that the observed difference could have occurred because of sampling and measurement error. When comparing jurisdictions, extensive use is made of this type of test to reduce the likelihood that any spurious differences due to sampling and measurement error be interpreted as real.
Only statistically significant differences at the 0.05 level are noted in this report, unless otherwise stated. This means that the 95 percent confidence intervals for the averages being compared do not overlap. Due to rounding error, some nonoverlapping confidence intervals share an upper or lower limit. All statistical differences are based on un-rounded data.
Data estimates, including mean scores, proportions and standard errors, are generally rounded to one decimal place. Therefore, even if the value (0.0) is shown for standard errors, this does necessarily imply that the standard error is zero, but that it is smaller than 0.05.
Please also note there may be inconsistencies between the numbers in the tables, charts and text of this report. Numbers presented in tables have been rounded up to or down at the nearest first decimal using data up to two decimal places. Whereas, the numbers shown in the charts have been rounded to the nearest whole number using data at one decimal place. Finally, the numbers presented in this report have been rounded up or down to the nearest whole number based on data up to two decimal places.
Example of inconsistencies between text, charts and tables:
Estimation with two decimals places is 4.48;
Value presented in the tables is 4.5;
Value presented in the charts is 5;
Value presented in the text is 4.
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Means and confidence intervals
Many figures in this report show the population mean scores surrounded by the 95th percent confidence intervals, and the scores at the 5th, 25th, 75th, and 95th percentiles for the domains of literacy and numeracy. The intervals at both ends of the proficiency continuum display the scores for the least and most proficient respondents — the 5th percentile to the lower bound of the distribution and the 95th to the upper bound of the distribution. The middle bar shows the mean and the confidence interval, within which the actual population average is assumed to fall. Moreover, these plots display the width of the distribution of the proficiency. A smaller spread of scores indicates fewer skill differences; a larger spread indicates more skill differences between the higher and lower performers.
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