DS 1000 Chapters 1–6 Practice Questions & Answers

Comprehensive vocabulary study flashcards covering all 236 practice questions and answers across Chapters 1 to 6 of DS 1000.

Data science

An interdisciplinary field that collects, stores, processes, analyzes, and visualizes data.

Individual

The object described by a set of data, such as a patient, student, or household.

Variable

A characteristic recorded for each individual, such as age, sex, height, or GPA.

Categorical variable

A variable that places individuals into groups or categories; arithmetic on its labels does not make sense.

Quantitative variable

A variable whose values are numerical measurements or counts for which arithmetic makes sense.

Classification of age, eye colour, height, number of siblings, and favourite subject

Age, height, and number of siblings are quantitative; eye colour and favourite subject are categorical.

Exploratory data analysis (EDA)

The process of examining data to describe its main features, usually beginning with graphs and then adding numerical summaries.

Distribution

Describes the values a variable takes and how often those values occur.

Categorical distribution

Shows the categories and the count or percentage of individuals in each category.

Symmetric distribution

A distribution whose left and right sides are approximately mirror images.

Right-skewed distribution

A distribution with a longer tail extending to the right; most observations tend to be on the lower side.

Left-skewed distribution

A distribution with a longer tail extending to the left; most observations tend to be on the higher side.

Uniform distribution

A distribution in which values or intervals occur at roughly similar frequencies.

Bimodal distribution

A distribution with two distinct peaks or modes.

Outlier

An observation that lies unusually far from the rest of the data.

Pie chart

A display used for showing a categorical distribution; each slice represents a category's share of the total.

Pie chart percentage sum

$100\%$, because the slices represent the whole group.

Bar graph

A display used for categorical data, with one bar per category and bar height showing its count or percentage.

Histogram

A display that groups numerical values into intervals (bins), where bar height represents the number or proportion of observations in each interval.

Difference between a histogram and a bar graph

A histogram displays quantitative intervals and its bars touch; a bar graph displays separate categories and bars are separated.

Features to examine in a histogram

Overall pattern, shape, centre, spread/variability, and unusual features such as outliers.

Stem-and-leaf plot

A display that separates each observation into a stem (leading digit or digits) and a leaf (final digit), preserving individual values.

Reason for ordering leaves in a stem plot

Ordering leaves makes the distribution easier to read and interpret.

Splitting stems

A method used to create more rows when a stem has too many leaves or the plot looks too compressed.

Time plot

A graph showing how a quantitative variable changes over time.

Axes of a time plot

Time is on the horizontal ($x$) axis; the measured variable is on the vertical ($y$) axis.

Seasonal variations

Patterns that repeat at regular, known intervals, such as higher ice-cream sales each summer.

Variable type of heart rate

Quantitative, because it is a numerical measurement.

Graphs for patient heart rates

A histogram or stem-and-leaf plot; a stem plot is useful when retaining the individual observations matters.

Graph for monthly store visits over three years

A time plot, because it shows change over time.

Mean

The arithmetic average: add all observations and divide by the number of observations.

Mean formula

$$\text{Mean} = \frac{\text{sum of observations}}{n}$$

Median

The middle observation after the data are arranged from smallest to largest.

Finding the median with an odd number of observations

Take the single middle value in the ordered list.

Finding the median with an even number of observations

Average the two middle values in the ordered list.

Reason for ordering data before finding the median

The middle position can only be identified after the values are sorted.

Resistant measure of centre

The median; extreme values usually have less effect on its position.

Sensitivity of the mean to extreme values

Every value contributes to the sum, so a very large or small value can pull the average.

Mean and median in a symmetric distribution

They are generally equal or close together.

Mean position in a skewed distribution

The mean is usually pulled farther into the long tail than the median.

Preferred centre measure for skewed data with outliers

The median, because it is resistant to extreme observations.

Mean of 4, 6, 8, 10, 12

$$\frac{40}{5} = 8$$

Median of 3, 5, 7, 9, 11

$$7$$

Median of 2, 4, 6, 8, 10, 12

$$\frac{6 + 8}{2} = 7$$

Mean and median of 5, 8, 10, 12, 15, 20, 25

$\text{Mean} = \frac{95}{7} \approx 13.57$; $\text{median} = 12$.

Mean and median of 65, 70, 75, 80, 90

$\text{Mean} = 76$; $\text{median} = 75$.

Mean and median of 60, 65, 70, 75, 100

$\text{Mean} = 74$; $\text{median} = 70$. The mean is pulled upward more by $100$.

Quartiles

Values that divide ordered data into four approximately equal parts.

Q1 (First quartile)

The median of the lower half of the ordered data.

Q2 (Second quartile)

The second quartile, which is the overall median.

Q3 (Third quartile)

The median of the upper half of the ordered data.

Percentile

The $p\text{th}$ percentile is a value with approximately $p\%$ of observations at or below it.

25th percentile

$Q_1$; about $25\%$ of observations are at or below it.

50th percentile

The median; about half the observations are at or below it.

75th percentile

$Q_3$; about $75\%$ of observations are at or below it.

Median-of-halves method

Order the data, find the overall median, then find the median of the lower half for $Q_1$ and upper half for $Q_3$. Follow course convention for including the median when $n$ is odd.

Five-number summary for 1, 2, 5, 9, 12, 15, 17, 21, 23, 25

$\text{Min} = 1$, $Q_1 = 3.5$, $\text{median} = 13.5$, $Q_3 = 22$, $\text{max} = 25$, using the median-of-halves method.

Five-number summary for 1, 2, 5, 9, 12, 15, 17, 21, 23, 25, 29

$\text{Min} = 1$, $Q_1 = 5$, $\text{median} = 15$, $Q_3 = 23$, $\text{max} = 29$, using the median-of-halves method.

Variability

Describes how spread out the observations are.

Interquartile range (IQR)

The spread of the middle $50\%$ of observations: $\text{IQR} = Q_3 - Q_1$.

Large IQR meaning

The middle half of the data is relatively spread out.

Small IQR meaning

The middle half of the data is relatively clustered.

1.5 × IQR rule

A value is flagged as a potential outlier if it is below $Q_1 - 1.5(\text{IQR})$ or above $Q_3 + 1.5(\text{IQR})$.

Outlier fences

$\text{Lower fence} = Q_1 - 1.5(\text{IQR})$; $\text{upper fence} = Q_3 + 1.5(\text{IQR})$.

Boxplot

A graph based on the five-number summary; the box spans $Q_1$ to $Q_3$ and the line inside marks the median.

Length of a box in a boxplot

The $\text{IQR}$, or spread of the middle $50\%$ of observations.

Standard deviation

A measure of the typical distance of observations from the mean.

Variance

The average squared deviation from the sample mean, using $n - 1$ for a sample variance.

Calculation of sample standard deviation

Find the mean; subtract it from each value; square deviations; add them; divide by $n - 1$ for variance; take the square root.

Reason for squaring deviations

Positive and negative deviations would otherwise cancel; squaring also gives more weight to larger deviations.

Reason for dividing by n − 1 in sample variance

It is the sample-variance adjustment used to estimate population variability.

Sign of standard deviation

It cannot be negative; it is the square root of a nonnegative variance.

Standard deviation equal to 0 (SD = 0)

There is no variability; all observations have the same value.

Units of variance and standard deviation

Variance is in squared units; standard deviation is in the original units.

Summary statistics for 90, 90, 100, 110, 110

$\text{Mean} = 100$. Deviations: $-10, -10, 0, 10, 10$. Squared deviations sum to $400$. $\text{Sample variance} = \frac{400}{5-1} = 100$; $\text{sample SD} = 10$.

Features to report when describing a distribution

Shape, centre, and variability/spread; also mention unusual features or outliers.

x-bar (x̄)

The sample mean.

µ

The mean of the population distribution.

s

The sample standard deviation.

σ

The standard deviation of the population distribution.

Density curve

A smooth curve used to describe the overall pattern of a distribution when many observations create a regular pattern.

Key properties of a density curve

It stays on or above the horizontal axis, and the total area underneath it equals $1$.

Area under a density curve

The proportion or probability of observations falling within the corresponding range of values.

Median on a density curve

The equal-areas point, dividing the area under the curve into two halves.

Mean on a density curve

The balance point of the curve, like the point where a solid cut-out would balance.

Mean and median in a symmetric density curve

Located at the centre; they are equal.

Mean in a skewed density curve

It is pulled toward the long tail, away from the median.

Normal distribution

A symmetric, bell-shaped density curve described by its mean and standard deviation.

Gaussian curve

Another name for a normal curve.

Standard normal distribution

The normal distribution with mean $0$ and standard deviation $1$, written $N(0, 1)$.

Effect of changing µ on a normal curve

Shifts the curve horizontally without changing its spread.

Effect of changing σ on a normal curve

Changes the spread: larger $\sigma$ makes it wider; smaller $\sigma$ makes it narrower.

68–95–99.7 rule

About $68\%$ lie within $1\text{ SD}$ of the mean, $95\%$ within $2\text{ SDs}$, and $99.7\%$ within $3\text{ SDs}$.

68% interval for mean 50 and SD 5

$45\text{ to }55$.

95% interval for mean 50 and SD 5

$40\text{ to }60$.

99.7% interval for mean 50 and SD 5

$35\text{ to }65$.

Percentage between 70 and 90 (mean 80, SD 10)

About $68\%$.

Percentage between 60 and 100 (mean 80, SD 10)

About $95\%$.

Percentage between 50 and 110 (mean 80, SD 10)

About $99.7\%$.

Standardizing

Converting a value to a number of standard deviations from the mean.