Math Review Lecture 4

1
Math ReviewLecture 4

• Evans School of Public Affairs
• University of Washington

2
Lecture 4 Describing Data

• Describing Data
• The average (arithmetic mean) of a group of
  numbers is defined as the sum of the values
  divided by the number of values.
• Average
• Example Heather buys three items costing 2.00,
  0.75, and 0.25. What is the average price?

3
Lecture 4 Describing Data

• If you know the average of a group of numbers and
  the number of numbers in the group, you can find
  the sum of the numbers. Its as if all the
  numbers in the group have the average value.
• Example The average daily temperature for the
  first week in January was 31 degrees. If the
  average temperature for the first six days was 30
  degrees, what was the temperature on the seventh
  day?
• The sum for all 7 days 31 7 217 degrees.
  The sum of the first six days 30 6 180
  degrees. The temperature on the seventh day 217
  - 180 37 degrees.

4
Lecture 4 Describing Data

• For evenly spaced numbers, the average is the
  middle value.
• Example 1 The average of consecutive integers 6,
  7, and 8 is 7.
• Example 2 The average of 5, 10, 15, and 20 is
  12.5 (midway between the middle values 10 and
  15).

5
Lecture 4 Describing Data

• For some people, it is useful to think of the
  average as the balanced value. That is, all the
  numbers below the average are less than the
  average by an amount that will balance out the
  amount that the numbers above the average are
  greater than the average.
• Example 1 the average of 3, 5, and 10 is 6. 3 is
  3 less than 6 and 5 is 1 less than 6. This, in
  total, is 4, which is the same as the amount that
  10 is greater than 6.
• Example 2 The average of 3, 4, 5, and x is 5.
  What is the value of x?

6
Lecture 4 Describing Data

• Median The median of a set of ascending numbers
  is the number that separates the higher half from
  the lower half.
• If there is an odd number of numbers, the median
  will be the same as the middle number (but not
  necessarily the average or the mode).
• If there is an even number of numbers, the median
  will be the average of the two numbers closest to
  the middle.
• Example The median of 4, 5, 7, 23, 5, 67, 10
  is 7.

7
Lecture 4 Describing Data

• Mode The mode is the most frequently occurring
  number in a list of numbers.
• Example The mode of 4, 5, 7, 23, 5, 67, 10 is
  5.
• Range The range is the difference between the
  largest number and the smallest number in a set
  of numbers. (Not the same as the range of a
  function.)
• Example In the set 4, 5, 7, 23, 5, 67, 10, the
  range is 63 because the greatest number, 67,
  minus the smallest, 4, equals 63.

8
Lecture 4 Describing Data

• A histogram shows the frequency of ranges of
  values of a variable (e.g. suppose 30 of people
  are 56 to 59).

9
Lecture 4 Distributions

• A probability distribution shows the range of
  values a variable can attain, and the probability
  of certain subsets of that range.
• Probability distributions are similar to
  histograms, but with infinitesimally small bin
  widths. Since a very small bin width has no area,
  we cannot talk about the probability of one
  particular value, but rather only about the
  probability of a range of values.

10
Lecture 4 Distributions

• A uniform distribution is one in which all
  intervals of the same width have equal
  probability. The graph is a rectangle.
• Example the chance of rolling a 2 or 3 on a die
  is the same as the chance of rolling a 4 or 5.

11
Lecture 4 Distributions

• A normal distribution is one in which the middle
  values are more likely than the extremes. In
  fact, 68 of observations are within one standard
  deviation (a measure of spread), and 95 of
  observations are within 2 std. deviations.

12
Lecture 4 Graphing Non-Linear Functions

• Non-linear functions are those that cannot be
  expressed in the form ymxb.
• Examples of non-linear functions include
• yx2
• yx(1/2)
• yln x
• y1/x
• yex

13
Lecture 4 Graphing Non-Linear Functions

• The graph of yx2 looks like this
• Whats the graph of y x24? And y2x2?

14
Lecture 4 Graphing Non-Linear Functions

• The basic form of a parabola is f(x) ax2bxc
• When a is positive, the curve opens upward. When
  a is negative, the curve opens downward.
• The y-intercept is c. The x-intercept is/are the
  0, 1, or 2 solution(s) to 0ax2bxc.
• For curves opening up, the vertex is the lowest
  point. For curves opening down, the vertex is the
  highest point.

15
Lecture 4 Graphing Non-Linear Functions

• To graph a quadratic equation
• Write the equation in standard form
• Determine whether the parabola opens up or down
• Plot the y-intercept (0,c)
• Use plug-and-chug to determine at least two other
  points (at least three in total) with which to
  sketch the graph. Making a chart of x and y
  values can be helpful.
• The coordinates of the vertex are

16
Lecture 4 Graphing Non-Linear Functions

• An asymptote is a line that x (or y) approaches
  as y (or x) goes to positive or negative
  infinity.
• One graph with asymptotes is y1/x.

17
Lecture 4 Non Constant Slope

• Non Constant Slope Unlike in graphs of linear
  equations, the slopes of quadratic and other
  non-linear equations are not constant.
• In the graph of yx2, for example, the slope of
  the line is 0 when x0. The slope is negative
  when xlt0 and positive when xgt0.

18
Lecture 4 Non Constant Slope

• A tangent line just touches a curve and has the
  same slope as the curve at that point.
• When the slope of the tangent is positive, the
  slope at the tangency point is positive.

19
Lecture 4 Non Constant Slope

• Recall that the slope of a straight line is
• This is only an approximate slope for a curve.
  The approximation gets more accurate with smaller
  x distances.
• The idea of a derivative in calculus is to reduce
  the x distance as much as possible in order to
  measure the slope at each value of x.

20
Lecture 4 Non Constant Slope

• The derivative of f(x)x2 is f(x)2x.
• (The power rule says if f(x) xn for any real
  number n, then f(x) nxn-1)
• This tells us that the slope of the parabola x2
  is always twice the value of x.
• When x0, the slope of x2 is 2(0)0.
• When x2, the slope of x2 is 2(2)4.
• When x-2, the slope of x2 is 2(-2)-4.