AP Statistics - Descriptive Statistics

Intro & Quantitative Data - TYPES OF DATA

• Quantitative: data in the form of numerical values

  • ex> height, weight

• Qualitative: data in the form of words, characteristics, etc.

  • ex> fav color, birthday month

Intro & Quantitative Data - TYPES OF GRAPHS

• For univariable (1 variable) data: bar graph, pie chart, histogram, line graph, stem + leaf plot, dot plot, box plot

• For bivariable (studies the relationship b/w 2 variables) data: scatter plot

Intro & Quantitative Data - Distribution

→ set of data that uses the frequency that each outcome occurs among all possibilities

• Measures of Central Tendency → where center of distribution of data lies

  • mean, median, mode

• Measures of Spread → amount of variation in distribution

  • range, IQR, standard deviation

• Shape of Distribution

Intro & Quantitative Data - Histogram

• title

• x-axis (+labels)

• y-axis (+labels)

• bars touch, measures a quantitative variable against frequency

Intro & Quantitative Data - Dot Plot

• title

• x-axis

• dots above corresponding values to represent frequency

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Wherever tail is…)

…pulls the mean up or down…

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Skew Right)

• Skew Right: most data on left

  • mean > med

  • high values have a big weight on mean

    • few data points to right pull mean up

  • tail w/ less data on right

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Skew Left)

• Skew Left: most data on right

  • mean < med

  • tail on left

  • few data points to left pull mean down

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Symmetric)

• mean = med

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Unimodal)

• “one mode”

  • One hump w/ highest frequency

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Uniform)

• frequencies are about the same

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Bimodal)

• (symmetric)

Intro & Quantitative Data - SHAPES OF DISTRIBUTIONS (Multimodal)

Intro & Quantitative Data - SYMBOLS: Population Mean

• μ (“mu”)

Intro & Quantitative Data - SYMBOLS: Sample Mean

• x̄ (x-bar)

  • x → any variable

Intro & Quantitative Data - SYMBOLS: Population Standard Deviation

• 𝛔 (sigma)

Intro & Quantitative Data - SYMBOLS: Population Variable

• 𝛔² (sigma squared)

Intro & Quantitative Data - SYMBOLS: Sample Standard Deviation

• s

Intro & Quantitative Data - SYMBOLS: Sample Variable

• s²

Intro & Quantitative Data - MEASURES OF CENTRAL TENDENCY

• Typically the mean best describes a distribution

• When outliers exist or a large skew, the median is best

  • outliers and skewedness affect the mean b/c the mean takes into account the weight of all values whereas the median does not

• Mode is used for qualitative data (you can’t find mean/median w/o #’s)

Intro & Quantitative Data - HISTOGRAM W/ CLASSES

• To create classes → Range / # of classes

  • (must be whole #, ALWAYS round up)

• Classes: use formula and add by class width for each class

• MP: (smaller number in class width + larger number in class width) / 2

  • x-axis

• Frequency: find how many numbers are present in the distribution in classes

  • Should add up to sample size!

• Relative Frequency: frequency/sample size

  • Y-AXIS

• Cumulative Relative Frequency: add up relative frequencies

  • Always ends at 1!

Intro & Quantitative Data - MEASURES OF SPREAD

• Range (max-min) = 29-5 = 24

  • *The range is 24 or the range is from 5 to 29

• IQR: interquartile range (Q₃ - Q₁)

• Standard deviation

Intro & Quantitative Data - BOX PLOTS

• List numbers in order

• Find MEDIAN

  • Median term # when listed in order

    • (n + 1) / 2

• Find Q₁

  • Median between median and minimum value

• Find Q₃

  • Median between median and maximum value

• 25% of the data is within each quartile

  • SIZE of quartile doesn’t matter (just indicates more or less spread)

• FOR OUTLIERS…

  • Solve for outliers

  • Make the maximum/minimum value the next highest number

Intro & Quantitative Data - 5 NUMBER SUMMARY

• Minimum

• Q₁

• Median

• Q₃

• Maximum

Intro & Quantitative Data - OGIVE: CUMULATIVE RELATIVE FREQUENCY GRAPH

• Plot points as a line

• x-axis: MP’s

• y-axis: Cumulative Relative Frequency

• *ogives are only interpreted to the left (‘this or less”)

• *to go from cumulative relative frequency to a box plot, estimate the quartiles (0%, 25%, 50%, 75%, 100%)

  • 0% → min

  • 25% → Q₁

  • 50% → Q₂

  • 75% → Q₃

  • 100% → max

Intro & Quantitative Data - STANDARD DEVIATION

→ the average distance each value lies from the mean

• Make a table with x, (x-x̄), & (x-x̄)²

• List data points under x column

• Do (x-x̄) under (x-x̄) column

  • Add up all the values

• Do (x-x̄)² under (x-x̄)² column

  • Add up all the values = TOTAL VARIABLE

• 𝛔² (population variable) = total variable / average variable

• 𝛔 (population standard deviation) = √𝛔² 

  • On average ____ stray ____ (𝛔) away from the mean.

Intro & Quantitative Data - FORMULAS FOR STANDARD DEVIATION

• For population:

  

• For sample:

  

Intro & Quantitative Data - CALCULATE OUTLIERS

• Rule is outliers fall outside of interval

  • [Q₁ - 1.5(IQR), Q₃ + 1.5(IQR)]

Intro & Quantitative Data - WRITE A FEW SENTENCES DESCRIBING THE DATA

1. center

2. spread

3. shape

4. unusual features (outliers, gaps, clusters)

5. MUST be in context ⭐

Describing Qualitative Data - BAR CHART

• x-axis

• y-axis: frequency

• bars DO NOT touch

Describing Qualitative Data - PARETO CHART

• x-axis

• y-axis: Frequency

• bars DO NOT touch

• *Bars in descending order, highlights the mode

Describing Qualitative Data - PIE CHART

• percentage = relative frequency

• # of people = frequency

Describing Qualitative Data - SEGMENTED BAR GRAPH

• Make a table with RELATIVE FREQUENCY & CUMULATIVE RELATIVE FREQUENCY

  • Add relative frequencies before value to get cumulative relative frequency

• x-axis: One Bar

• y-axis: Cumulative Relative Frequency

• label segments of bar

• *break messes with scale… can make relative frequency look smaller than it is

Describing Qualitative Data - CONTINGENCY TABLE

• 2 variables

• ….of the… = denominator

Comparing Distributions

• Include a discussion of center, spread and shape using context and comparative statements. Include approximate values/ranges when possible.

Comparing Distributions - Comparative Statements

• Comparative Statements: greater than, higher, less than, lower, equal, etc. (except shape)

  • Use “whereas” only for shape

• List:

  • mean

  • standard deviation

  • sample size

  • minimum value

  • Q₁

  • median

  • Q₃

  • maximum value

  • outliers

Introduction to Normal Distributions - normal distribution

• a bell-shaped frequency distribution curve. Most of the data values in a normal distribution tend to cluster around the mean.

  • → the further away a data point is from the mean, the less likely it is to happen

Introduction to Normal Distributions - Characteristics

• unimodal (one mode, one peak), symmetric (right side mirrors left side), asymptotic (approach, but never touch x-axis), mean = median = mode (center = peak → 50% data below mean, 50% data above mean)

Introduction to Normal Distributions - What does the NORMAL MODEL look like?

• x-axis: mean @ center + standard deviations away

• curve with asymptotic ends

Names for Normal Distributions

• One of the most important examples of a continuous probability distribution is the normal distribution. The graph is usually called normal, bell-shaped or Gaussian curve.

Properties of Normal Distributions - Area Under the Curve

• Total area under the curve is always equal to one.

• The portion of the area under the curve above a given interval represents the probability that a measurement will lie in that interval.

  • area under curve = probability

Properties of Normal Distributions - EMPIRICAL FORMULA

• The Empirical Rule can be applied for any normal distribution which says:

  

  • → about 68% of data lies within 1 std. dev. of mean

  • →about 95% of data lies within 2 std. dev. of mean

  • → about 99.7% of data lies within 3 std. dev. of mean

• The Empirical Rule can be used to find different percentiles.

• *MAKE SURE TO INCLUDE ABOUT WHEN ANSWERING QUESTIONS

Properties of Normal Distributions - Normal distributions vary from one another in two ways: the mean may be located anywhere on the x axis and the bell shape may be more or less spread according to the size of the standard deviation. It would be difficult to compute the area under the curve for each different combination… Z SCORES

• → a z-score tells you exactly how many std. dev. a data value is above or below the mean

Properties of Normal Distributions - Z SCORES: How does standardizing affect the center, spread and shape of the distribution?

• When the data is converted to z scores the mean (center) becomes 0, the std. dev. becomes 1, shape remains the same

• z = (x - μ) / 𝛔 → standardized test statistic = (statistic - parameter) / std. Dev.

Properties of Normal Distributions - Z SCORES: We can use these z-scores to then…

… calculate probabilities using our z-score chart to determine the area under the curve that corresponds with each z-score.

Properties of Normal Distributions - Z SCORES: Find the specified areas!

• 4 decimal places b/c table uses 4 decimal places (for area under curve/probability)

  • Go to z-score table using probability notation → P(z </>/</> _)

    • For negative values (to the left), find z score & corresponding value.

    • For positive values (to the right), subtract corresponding values from 1.

    • For between, (larger value in table) - (smaller value in table).

Properties of Normal Distributions - Z SCORES: < / <

• < / < are the same!

Properties of Normal Distributions - ACCURACY

• ACCURACY: The normal model is not accurate if <3 std. dev. from the mean is negative.

  • Depends on the context of the problem…

  •  A normal model must be able to go 3 std. dev. in both directions.

Rescaling Data - How does shift affect mean + std. dev.?

• The mean increases or decreases by shift.

• The std. dev. stays the same (not affected)

Rescaling Data - How does multiplier affect mean + std. dev.?

• → multiplying (scaling) data

• Both mean + std. dev. get multiplied by the scalar value.

Rescaling Data - Adding a number to a distribution that is the same as the mean will…

• not change the mean as it is equal to the current mean

• decrease the std. dev. because there is less variability since we have added another value at the center

Rescaling Data - *when you convert units…

…nothing is actually changing (this applies to z-scores as well)

Rescaling Data - Turn rescaling into an algebraic expression!

• *substitute values you are looking for into expression… need to take into account which values are affected by shifts/multipliers.

• *data points/measures of center affected by both shifts + mult. while measures of spread are only affected by mult.