Unit One: Exploring One-Variable Data- essential knowledge

What does it mean when numbers are placed in context?

Numbers may convey meaningful information when placed in context.

What is a variable in statistics?

A variable is a characteristic that changes from one individual to another.

What is a categorical variable?

A categorical variable takes on values that are category names or group labels.

What is a quantitative variable?

A quantitative variable takes on numerical values for a measured or counted quantity.

What does a frequency table show?

A frequency table gives the number of cases falling into each category.

What is a relative frequency table?

A relative frequency table gives the proportion of cases falling into each category.

How are percentages, relative frequencies, and rates related to proportions?

Percentages, relative frequencies, and rates all provide the same information as proportions.

How can counts and relative frequencies be used in categorical data?

Counts and relative frequencies of categorical data reveal information that can be used to justify claims about the data in context.

What are bar charts used for?

Bar charts (or bar graphs) are used to display frequencies (counts) or relative frequencies (proportions) for categorical data.

What does the height or length of each bar in a bar graph represent?

The height or length of each bar corresponds to the number or proportion of observations in each category.

Are there additional ways to represent frequencies or relative frequencies for categorical data?

Yes, there are many additional ways to represent frequencies (counts) or relative frequencies (proportions) for categorical data.

What can graphical representations of a categorical variable reveal?

Graphical representations of a categorical variable reveal information that can be used to justify claims about the data in context.

How can frequency tables or bar graphs be used for comparison?

Frequency tables, bar graphs, or other representations can be used to compare two or more data sets in terms of the same categorical variable.

What is a discrete variable?

A discrete variable can take on a countable number of values, either finite or countably infinite.

What is a continuous variable?

A continuous variable can take on infinitely many values, with the possibility of determining another value between any two values.

What does the height of each bar in a histogram represent?

The height of each bar shows the number or proportion of observations in the interval corresponding to that bar.

How does altering interval widths affect a histogram?

Altering the interval widths can change the appearance of the histogram.

How is a stem and leaf plot structured?

In a stem and leaf plot, each data value is split into a “stem” (the first digit or digits) and a “leaf” (usually the last digit).

What does a dotplot show?

A dotplot represents each observation by a dot, with positions on the horizontal axis corresponding to data values, and similar values stacked on top of each other.

What does a cumulative graph represent?

A cumulative graph represents the number or proportion of a data set less than or equal to a given number.

Are there other ways to graphically represent quantitative data distributions?

Yes, there are many additional ways to graphically represent distributions of quantitative data.

What are key features to describe in a distribution of quantitative data?

Descriptions include shape, center, variability (spread), and unusual features like outliers, gaps, clusters, or multiple peaks.

What is an outlier in one-variable data?

An outlier is a data point that is unusually small or large relative to the rest of the data.

When is a distribution skewed to the right?

A distribution is skewed to the right (positively skewed) if the right tail is longer than the left.

When is a distribution skewed to the left?

A distribution is skewed to the left (negatively skewed) if the left tail is longer than the right.

What does it mean for a distribution to be symmetric?

A distribution is symmetric if the left half is a mirror image of the right half.

What is a unimodal distribution?

A unimodal distribution has one main peak.

What is a bimodal distribution?

A bimodal distribution has two prominent peaks.

What is an approximately uniform distribution?

An approximately uniform distribution has bars of similar height with no prominent peaks.

What is a gap in a distribution?

A gap is a region in a distribution between data values where there are no observed data.

What are clusters in a distribution?

Clusters are concentrations of data points usually separated by gaps.

What is the role of descriptive statistics?

Descriptive statistics do not attribute properties of a data set to a larger population but may provide the basis for conjectures for further testing.

What is a statistic in statistics?

A statistic is a numerical summary of sample data.

How is the mean calculated for a sample?

The mean is the sum of all data values divided by the number of values.

What is the median of a data set?

The median is the middle value when data are ordered. If there is an even number of data points, the median is the average of the two middle values.

What are the first and third quartiles (Q1 and Q3)?

Q1 is the median of the lower half of the ordered data, and Q3 is the median of the upper half.

What is the interpretation of the pth percentile?

The pth percentile is the value that has p% of the data less than or equal to it.

What are three common measures of variability?

Range, interquartile range (IQR), and standard deviation are commonly used measures of variability.

How is the range calculated?

The range is the difference between the maximum and minimum data values.

How is the interquartile range (IQR) calculated?

The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

What does the standard deviation measure?

The standard deviation measures the variability of a quantitative variable’s distribution.

What is a five-number summary?

A five-number summary consists of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum data values.

What is a boxplot?

A boxplot is a graphical representation of the five-number summary with whiskers extending to the most extreme non-outlier data points and outliers represented separately.

How does the mean compare to the median in different distributions?

In a symmetric distribution, the mean and median are close. If skewed right, the mean is usually to the right of the median; if skewed left, the mean is usually to the left of the median.

How can you compare two or more data sets using graphical representations?

Graphical representations like histograms or side-by-side boxplots can be used to compare center, variability, clusters, gaps, outliers, and more.

How can you compare data sets using numerical summaries?

Numerical summaries such as mean, standard deviation, and relative frequency can be used to compare different data sets.

What is a parameter?

A parameter is a numerical summary of a population.

What does a normal distribution look like?

A normal distribution is mound-shaped and symmetric.

What is the empirical rule for a normal distribution?

The empirical rule states that 68% of observations are within 1 standard deviation of the mean, 95% are within 2, and 99.7% are within 3 standard deviations.

How do you calculate a standardized score?

standardized score is calculated as (data value – mean) / (standard deviation), indicating how many standard deviations a data value is from the mean.

What is a z-score?

A z-score measures how many standard deviations a data value is from the mean.

How can technology be used in normal distributions?

Technology like calculators or software can be used to find proportions of data values in intervals of a normally distributed random variable.

How can you estimate population parameters given the area under a normal curve?

Technology can be used to estimate parameters for populations based on the area under a normal distribution curve.

How can percentiles and z-scores be used for comparisons?

Percentiles and z-scores compare the relative positions of points within a data set or between data sets.