Exploring One-Variable Data · ⁨استكشاف بيانات متغير واحد⁩

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.A: Identify questions to be answered, based on variation in one-variable data. [Skill 1.A]

• VAR-1.A.1 Numbers may convey meaningful information, when placed in context.

Enduring Understanding (VAR-1): Given that variation may be random or not, conclusions are uncertain.

Learning Objective VAR-1.B: Identify variables in a set of data. [Skill 2.A]

• VAR-1.B.1 A variable is a characteristic that changes from one individual to another.

Learning Objective VAR-1.C: Classify types of variables. [Skill 2.A]

• VAR-1.C.1 A categorical variable takes on values that are category names or group labels.
• VAR-1.C.2 A quantitative variable is one that takes on numerical values for a measured or counted quantity.
  • Illustrative examples for VAR-1.C:
    • Categorical variables:
      • Dominant hand
      • Age group (young or old)
      • Highest degree earned
    • Quantitative variables:
      • Age of a structure
      • Height of a child
      • Concentration of a sample

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.A: Represent categorical data using frequency or relative frequency tables. [Skill 2.B]

• UNC-1.A.1 A frequency table gives the number of cases falling into each category. A relative frequency table gives the proportion of cases falling into each category.

Learning Objective UNC-1.B: Describe categorical data represented in frequency or relative tables. [Skill 2.A]

• UNC-1.B.1 Percentages, relative frequencies, and rates all provide the same information as proportions.
• UNC-1.B.2 Counts and relative frequencies of categorical data reveal information that can be used to justify claims about the data in context.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.C: Represent categorical data graphically. [Skill 2.B]

• UNC-1.C.1 Bar charts (or bar graphs) are used to display frequencies (counts) or relative frequencies (proportions) for categorical data.
• UNC-1.C.2 The height or length of each bar in a bar graph corresponds to either the number or proportion of observations falling within each category.
• UNC-1.C.3 There are many additional ways to represent frequencies (counts) or relative frequencies (proportions) for categorical data.

Learning Objective UNC-1.D: Describe categorical data represented graphically. [Skill 2.A]

• UNC-1.D.1 Graphical representations of a categorical variable reveal information that can be used to justify claims about the data in context.

Learning Objective UNC-1.E: Compare multiple sets of categorical data. [Skill 2.D]

• UNC-1.E.1 Frequency tables, bar graphs, or other representations can be used to compare two or more data sets in terms of the same categorical variable.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.F: Classify types of quantitative variables. [Skill 2.A]

• UNC-1.F.1 A discrete variable can take on a countable number of values. The number of values may be finite or countably infinite, as with the counting numbers.
• UNC-1.F.2 A continuous variable can take on infinitely many values, but those values cannot be counted. No matter how small the interval between two values of a continuous variable, it is always possible to determine another value between them.
  • Illustrative examples for UNC-1.F:
    • A discrete variable:
      • Number of students in a class
    • A continuous variable:
      • Height of a child

Learning Objective UNC-1.G: Represent quantitative data graphically. [Skill 2.B]

• UNC-1.G.1 In a histogram, the height of each bar shows the number or proportion of observations that fall within the interval corresponding to that bar. Altering the interval widths can change the appearance of the histogram.
• UNC-1.G.2 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).
• UNC-1.G.3 A dotplot represents each observation by a dot, with the position on the horizontal axis corresponding to the data value of that observation, with nearly identical values stacked on top of each other.
• UNC-1.G.4 A cumulative graph represents the number or proportion of a data set less than or equal to a given number.
• UNC-1.G.5 There are many additional ways to graphically represent distributions of quantitative data.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.H: Describe the characteristics of quantitative data distributions. [Skill 2.A]

• UNC-1.H.1 Descriptions of the distribution of quantitative data include shape, center, and variability (spread), as well as any unusual features such as outliers, gaps, clusters, or multiple peaks.
• UNC-1.H.2 Outliers for one-variable data are data points that are unusually small or large relative to the rest of the data.
• UNC-1.H.3 A distribution is skewed to the right (positive skew) if the right tail is longer than the left. A distribution is skewed to the left (negative skew) if the left tail is longer than the right. A distribution is symmetric if the left half is the mirror image of the right half.
• UNC-1.H.4 Univariate graphs with one main peak are known as unimodal. Graphs with two prominent peaks are bimodal. A graph where each bar height is approximately the same (no prominent peaks) is approximately uniform.
• UNC-1.H.5 A gap is a region of a distribution between two data values where there are no observed data.
• UNC-1.H.6 Clusters are concentrations of data usually separated by gaps.
• UNC-1.H.7 Descriptive statistics does not attribute properties of a data set to a larger population, but may provide the basis for conjectures for subsequent testing.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.I: Calculate measures of center and position for quantitative data. [Skill 2.C]

• UNC-1.I.1 A statistic is a numerical summary of sample data.
• UNC-1.I.2 The mean is the sum of all the data values divided by the number of values. For a sample, the mean is denoted by $x$-bar: $\bar{x} = \dfrac{1}{n}\sum_{i=1}^{n} x_i$, where $x_i$ represents the $i^{\text{th}}$ data point in the sample and $n$ represents the number of data values in the sample.
• UNC-1.I.3 The median of a data set is the middle value when data are ordered. When the number of data points is even, the median can take on any value between the two middle values. In AP Statistics, the most commonly used value for the median of a data set with an even number of values is the average of the two middle values.
• UNC-1.I.4 The first quartile, Q1, is the median of the half of the ordered data set from the minimum to the position of the median. The third quartile, Q3, is the median of the half of the ordered data set from the position of the median to the maximum. Q1 and Q3 form the boundaries for the middle 50% of values in an ordered data set.
• UNC-1.I.5 The $p^{\text{th}}$ percentile is interpreted as the value that has $p\%$ of the data less than or equal to it.

Learning Objective UNC-1.J: Calculate measures of variability for quantitative data. [Skill 2.C]

• UNC-1.J.1 Three commonly used measures of variability (or spread) in a distribution are the range, interquartile range, and standard deviation.
• UNC-1.J.2 The range is defined as the difference between the maximum data value and the minimum data value. The interquartile range (IQR) is defined as the difference between the third and first quartiles: $Q3 - Q1$. Both the range and the interquartile range are possible ways of measuring variability of the distribution of a quantitative variable.
• UNC-1.J.3 Standard deviation is a way to measure variability of the distribution of a quantitative variable. For a sample, the standard deviation is denoted by $s$: $s_x = \sqrt{\dfrac{1}{n-1}\sum(x_i - \bar{x})^2}$. The square of the sample standard deviation, $s^2$, is called the sample variance.
• UNC-1.J.4 Changing units of measurement affects the values of the calculated statistics.

Learning Objective UNC-1.K: Explain the selection of a particular measure of center and/or variability for describing a set of quantitative data. [Skill 4.B]

• UNC-1.K.1 There are many methods for determining outliers. Two methods frequently used in this course are:
  • UNC-1.K.1.i An outlier is a value greater than $1.5 \times \text{IQR}$ above the third quartile or more than $1.5 \times \text{IQR}$ below the first quartile.
  • UNC-1.K.1.ii An outlier is a value located 2 or more standard deviations above, or below, the mean.
• UNC-1.K.2 The mean, standard deviation, and range are considered nonresistant (or non-robust) because they are influenced by outliers. The median and IQR are considered resistant (or robust), because outliers do not greatly (if at all) affect their value.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.L: Represent summary statistics for quantitative data graphically. [Skill 2.B]

• UNC-1.L.1 Taken together, the minimum data value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum data value make up the five-number summary.
• UNC-1.L.2 A boxplot is a graphical representation of the five-number summary (minimum, first quartile, median, third quartile, maximum). The box represents the middle 50% of data, with a line at the median and the ends of the box corresponding to the quartiles. Lines ("whiskers") extend from the quartiles to the most extreme point that is not an outlier, and outliers are indicated by their own symbol beyond this.

Learning Objective UNC-1.M: Describe summary statistics of quantitative data represented graphically. [Skill 2.A]

• UNC-1.M.1 Summary statistics of quantitative data, or of sets of quantitative data, can be used to justify claims about the data in context.
• UNC-1.M.2 If a distribution is relatively symmetric, then the mean and median are relatively close to one another. If a distribution is skewed right, then the mean is usually to the right of the median. If the distribution is skewed left, then the mean is usually to the left of the median.

Enduring Understanding (UNC-1): Graphical representations and statistics allow us to identify and represent key features of data.

Learning Objective UNC-1.N: Compare graphical representations for multiple sets of quantitative data. [Skill 2.D]

• UNC-1.N.1 Any of the graphical representations, e.g., histograms, side-by-side boxplots, etc., can be used to compare two or more independent samples on center, variability, clusters, gaps, outliers, and other features.

Learning Objective UNC-1.O: Compare summary statistics for multiple sets of quantitative data. [Skill 2.D]

• UNC-1.O.1 Any of the numerical summaries (e.g., mean, standard deviation, relative frequency, etc.) can be used to compare two or more independent samples.