Summary Statistics

One of the main purposes of statistics is to draw conclusions about a (usually large) population from a (usually small) sample of observed values.

Population – The collection of all individuals, items or data under consideration in a statistical study.

Sample – That part of the population from which information is collected.

Descriptive Statistics -

Methods of organizing and summarizing information in a clear and effective way.

Inferential Statistics -

Methods of drawing conclusions about a population based on information obtained from a sample of the population.

Parameter –

Statistic -

A descriptive measure for a population.

A descriptive measure for a sample.

Summary statistics for Discrete Data

Median:The middle value, after the observations have been arranged in order of magnitude. If the total number of values is even, then the median would be halfway between the middle two values.

Mode: The set of data which occur most frequently.

For grouped data, the term modal class is used.

Mean:

The Mean of the Population is generally unknown.

The Sample Mean serves as an unbiased estimate of the Population Mean.

Population Mean Sample Mean

1

k

i ii

f xx

n

1

k

i ii

f x

n

The shoe sizes of the members of a football team are:

10, 10, 8, 11, 10, 9, 9, 10, 11, 9, 10

Determine the mean, median and mode.

Answer: 9.73, 10, 10

Mean of Grouped Data

1

k

j jj

f xx

n where each value of x is the midpoint

of each class.

Each day, x, the number of diners in a restaurant was recorded and the following grouped frequency table was obtained.

Using the above grouped data, find an estimate of the mean.

x 16-20 21-25 26-30 31-35 36-40

Number of Diners

67 74 38 39 42

Answer: 26.4

Find the mean for the following set of data:

Interval Frequency20-24 125-29 630-34 1035-39 240-44 1

Answer: 31

A test marked out of 100 is written by 800 students. The cumulative frequency graph for the marks is given below.

(a) Write down the number of students who scored 40 marks or less on the test. (b) The middle 50 % of test results lie between marks a and b, where a < b. Find a and b.

Answers: a) 100 b) a = 75, b = 55

SPEC06/HL1/3

The heights of 60 children entering a school were measured. The following cumulative frequency graph illustrates the data obtained.

Estimate

(a) the median height;

(b) the mean height.Answers: Median = 1.04, Mean = 1.05

M04/HL1/15

The box and whisker plots shown represents the heights of female students and the heights of male students at a certain school.

(a) What percentage of female students are shorter than any male students?

(b) What percentage of male students are shorter than some female students?

(c) From the diagram, estimate the mean height of the male students.

Answer: (a) 25% (b) 75% (c) 172 cm N00/HL1/4

A die is rolled twenty times with the following results:

Given that the mean is 3.6, find the values of a and b.

Outcome 1 2 3 4 5 6

Frequency 2 4 A 7 2 B

Answer: a = 2, b = 3

Variance

The Variance tells us about how far the data lies from the Mean.

Population Variance

2 2

2 21 1

( )k k

i i i ii i

f x f x

n n

The population variance is generally unknown. The Sample Variance is slightly skewed and is biased. That is to say that the sample variance is different from the Population Variance.

Sample Variance

Unbiased Estimate of the Population Variance

2 2

22 1 1

( )k k

i i i ii i

n

f x x f xs x

n n

2 2

22 2 1 11

( )

1 1 1 1

k k

i i i ii i

n n

f x x f xn ns s xn n n n

Therefore, we also have an unbiased estimate of the Population Variance.

Standard Deviation:

The Standard Deviation is the square root of the variance.

Population Standard Deviation

Sample Standard Deviation

2

1

( )k

i ii

f x

n

2

1

( )k

i ii

n

f x xs

n

The nine planets of the solar system have approximate equatorial diameters (in thousands of km) as follows:

4.9, 12.1, 12.8, 6.8, 142.8, 120.0, 52.4, 49.5, 2.5

Determine the standard deviation of these diameters.

Answer: 49.7

Find the standard deviation for the following set of data:

Interval Frequency1-7 4

8-14 515-21 1022-28 6

Answer: 7.01

A grouped data for the number of days to maturity for 40 short-term investments is given below. Compute the sample mean and standard deviation.

Days to maturity Frequency

30-39 3

40-49 1

50-59 8

60-69 10

70-79 7

80-89 7

90-99 4

40

Answer:

Mean = 68.0 days

Standard Deviation = 16.2 days

Find the unbiased standard deviation:

Answer:

Unbiased

Standard Deviation = 16.4 days

A machine tests the distance, w, measured in thousands of km, that car tires travel before the tire wear reaches a critical amount. For a random sample of tires, the results are summarizes below:

12

23

48

15

3

0 25w

25 30w

30 35w

35 45w

45 60w

(a) Find the grouped mean for this data

(b) Find the unbiased estimate of the variance based on the grouped data.

Answers:

30700 km

70900

A sample of 70 batteries were tested to how long they last. The results were:

Determine:

(a) the sample standard deviation

(b) an unbiased estimate of the standard deviation from which this sample is taken.

Answers:

(a) 21.4 hours

(b) 21.6 hours

M00/HL1/4

For a set of 9 numbers and . Find the mean of the numbers.

2( ) 60x x 2 285x

Answer: mean = 5

For a given frequency distribution:2 2( ) 182.3, 1025, 30f x x fx f

find: .fx

Answer: 159

A teacher drives to school. She records the time taken on each of 20 randomly chosen days. She finds that

where denotes the time, in minutes, taken on the ith day.

Calculate an unbiased estimate of

(a) the mean time taken to drive to school;

(b) the variance of the time taken to drive to school.

20 202

1 1

626 and 19780.8i ii i

x x

ix

Answers:

(a) 31.3

(b) 9.84

M03/HL1/19

Chebychev’s Rule

Property 1: At least 75% of the data lie within two standard deviations to either side of the mean.

Property 2: At least 89% of the data lie within three standard deviations to either side of the mean.

Property 3: In general, for any number k > 1, at least

of the data lies within k standard deviations to either side of the mean.

211k

Z score: The z-score for a data value is the number of standard deviations that the data value is away from the mean.

Sample z-score

Population z-score

x xzs

xz

It is known that a coffee machine dispenses an average of 6 fluid ounces of coffee with a standard deviation of 0.2 fluid ounces. A cup of coffee dispensed from the machine is found to contain 7.1 fluid ounces of coffee. Determine and interpret the z-score for this cup of coffee. Does this cup of coffee contain an unusually large amount?

Answer:

The z-score is 5.5, or 5.5 standard deviations above the mean.

This cup contains more coffee than 96.7% of all cups dispensed, it is a large amount.

Two major topics of inferential statistics:

1. Using the sample mean to make inferences about the population mean.

2. Using the sample standard deviation to make inferences about the population standard deviation.