Transcript Math Concepts

Math Concepts
Biological and
Physical Sciences
Objective
Understand basic math and statistic
concepts
Overview
Mean, Median and Mode
Proportions, Ratios and Rate
Summation
Percentiles
Graphs
Mean (Average)
A mean is the sum of all observed values
divided by the count of the total number of
observations.
What is the mean of 3, 5, 8, 10?
(3+5+8+10)/4 = 6.5.
Median
•
•
The median of a set of numbers can be found by
arranging all the numbers from lowest value to highest
value and picking the middle one.
What is the median of 3, 2, 4, 5, 7?
– Arrange in ascending order first: 2, 3, 4, 5, 7. The
middle number is 4.
• What is the median of 2, 1, 1, 3?
– Arrange in ascending order first: 1, 1, 2, 3. Since
there are four numbers, the middle number is
between the second and third number, or 1.5.
Mode
The mode is the number that occurs most
often in a set of data.
What is the mode of 1, 1, 2, 3?
1, since it occurs most often.
What is the mode of 1, 2, 3, 4?
There is no mode since all numbers
occur with same frequency.
Questions
In a class of six students, the students’
ages were: 19, 21, 22, 32, 30, 20.
What is the mean age of the students?
What is the median age of the
students?
What is the mode of ages?
Answers
Mean = (19+21+22+32+30+20)/6 = 24.
Arrange in ascending order first: 19, 20,
21, 22, 30, 32. Median is between the
third and fourth number, or 21.5.
Since no number occurs more frequently
than others, there is no mode.
Range
The range of a set of data is the difference
between its largest and smallest values.
What is the range of 2, 3, 1, 0, 8?
Arrange in ascending order first: 0, 1, 2,
3, 8. Range is 8-0 = 8.
Question
What is the range of 43, 23, 35, 24?
Answer
Arrange in ascending order first: 23, 24,
35, 43. Range is 43-23 = 20.
Proportion
•
A proportion is the count of observations with
a given value or set of values divided by the
count of the total number of observations.
•
Suppose you have a sample of five numbers:
3, 4, 6, 8, 10. What is the proportion of
numbers less than 7?
– Since you have three numbers less than 7
out of five numbers, 3/5 = 0.6.
Question
In a class of six students, the students’
ages were: 19, 21, 22, 32, 30, 20.
What is the proportion of students who
are younger than 23?
Answer
Proportion of the students younger than
23 = 4/6 = 0.67.
Ratio
•
A ratio is any division of one number by
another; the numerator and denominator do
not have to be mutually exclusive.
•
In a basket, there are six apples and two
oranges. What is the ratio of apples to
oranges?
– 6/2 or 3. In this example, the apples have
no relationship with the oranges.
Proportion (Another Way
to Look At It)
•
A proportion is a ratio in which those in the
numerator are also in the denominator.
•
In a basket, there are six green apples and
two red apples. What proportion of the
apples are green?
– 6/8 or 3/4 or 0.75. In this example, the
numerator (six green apples) is included in
the denominator (a total of eight apples).
Rate
•
A rate is a ratio in which those in the numerator are
also in the denominator, and those in the denominator
are "at risk" of being in the numerator.
•
In a basket, there are six green apples and two red
apples. Furthermore, two of the six green apples are
spoiled. What is the rate of spoiled apples in the
basket?
– 2/8 or 1/4 or 0.25. In this example, the numerator
(two spoiled green apples) is included in the
denominator (a total of eight apples). Furthermore,
those in the denominator (all eight apples) are at
risk of being in the numerator (spoiled apples).
Questions
In a basket, there are three bananas, two
green apples, and five red apples. Two of
the red apples are spoiled.
What is the ratio of bananas to red
apples?
What proportion of the apples are red?
What is the rate of spoiled apples in the
basket?
Answers
Ratio of bananas to red apples = 3/5 or
0.6.
Proportion of the red apples = 5/7 or 0.71.
Rate of spoiled apples = 2/7 or 0.29.
Summation
•
Summation sign is used to denote addition
of a set of numbers.
•
Instead of writing 1+2+3+4, you can write:
4
∑i
i=1
Questions
4
•
∑i=
i=2
4
•
∑ n2 =
n=1
Answers
4
•
∑ i = 2+3+4 = 9
i=2
4
•
∑ n2 = (1)2+(2)2+(3)2+(4)2 = 30
n=1
Percentiles
•
The xth percentile of a distribution is the value
such that x percent of the observations fall at or
below it.
•
To find the position of the first and third quartiles
(or 25th or 75th percentiles), you use the formula:
– Position of first quartile = (n+1)/4, where n is
the number of observations.
– Position of third quartile = 3*(n+1)/4, where n is
the number of observations.
Percentiles (continued)
•
For 2, 6, 4, 10, find the position of the first and
third quartiles.
– Arrange in ascending order first: 2, 4, 6, 10.
• Position of first quartile = (4+1)/4 = 1.25.
• Position of third quartile = 3*(4+1)/4 = 3.75.
• This means that the position of first quartile lies
one-fourth of the way between the first and
second observations and the position of third
quartile lies three-fourths of the way between the
third and fourth observations.
Percentiles (continued)
The value of first quartile then is onefourth of the way between the first (2) and
second (4) observations, or 2+ ¼ (4-2) =
2.5. The value of third quartile similarly is
three-fourths of the way between the third
(6) and fourth (10) observations, or 6+ ¾
(10-6) = 9.
This also means that 25% of values fall
below 2.5 (2) and 75% of values fall below
9 (2, 4, 6).
Questions
For 1, 2, 3, 4, 5, 6, find the values of first
and third quartiles. Explain in terms of
percentiles.
Answers
•
Since there are six observations, n = 6.
Position of first quartile = (6+1)/4 = 1.75.
Position of third quartile = 3*(6+1)/4 = 5.25.
The value of first quartile is three-fourths of the way
between the first (1) and second (2) observations, or
1+ ¾ (2-1) = 1.75. The value of third quartile is onefourth of the way between the fifth (5) and sixth (6)
observations, or 5+ ¼ (6-5) = 5.25. This means 25%
of values fall below 1.75 and 75% of values fall below
5.25.
Graph
Graphs are used to visually summarize
data. Independent variables are on the xaxis and dependent variables are on the
y-axis.
Graph (continued)
Suppose you measured someone’s age and
corresponding height:
Age
Height (cm)
2
40
5
50
8
75
10
90
12
125
15
160
17
175
20
180
23
180
Line Graph
y-axis
Height
200
150
100
50
x-axis
5
10
15
Age
20
25
Another Example of
Bar Graph
Questions??