Finding the precise median for a continuous variable

Quantitative methods ECON 530

Session 1 Notes

The course Quantitative methods deals with the application of statistics and mathematics methods to the estimation of economic problems. During the first half of the semester we will deal with the application of statistical tools and techniques in economics. In the second half we will deal with Mathematical models.

Statistics is the science which deals with the method of collecting, classifying, comparing and interpreting numerical data collected to throw some light on any sphere of enquiry. The definition points out the various statistical methods or features of science of statistics. They are collection of data, classification of data, presentation of data and analysis and interpretation of these presented data. Some of the important functions of statistics include the following:

1. It simplifies the complexity: In our studies we collect huge facts and figures. They can’t be easily understood. Statistical methods make these large numbers of facts easily and readily understandable. In statistics there are methods like – graphs and diagrams, classifications, average etc which render complex data very simple.

2. It presents fact in a definite Form: One of the most important functions of statistics is to present general statements in a precise and definite form.

3. It facilitates comparison: Unless figures are comparing with other of the same kind, they are often devoid of any meaning. When we say that the price of a commodity has increased very much, the statement does not make the positions very clear. But when we say that last year price was 10 but now it is 11 the comparison becomes easy. Statistics provides a number of suitable methods like ratios, percentages, averages etc for comparison.

4. It helps in formulating and testing of hypothesis: Statistical methods are extremely helpful in formulating and testing hypothesis and to develop new theories.

5. It helps in prediction: Plans and policies of organizations are invariably formulated well in advance of the time of their implementation. Knowledge of future tendencies is very helpful in framing suitable policies and plans. Statistical methods provide helpful means of forecasting future events.

6. Formulation of policies: Statistics provide the basic material for framing suitable policies. The statistical tools like collection of data help much in this regard.

Statistics and Economics:

As you well know economics is concerned with production and distribution of wealth as well as with the complex institutional set-up connected with the consumption, saving and investment of income. Statistical data and statistical methods are of immense help in the proper understanding of the economic problems and in the formulation of economic policies. In fact these are the tools and appliances of an economist’s laboratory. In the field of economics it is almost impossible to find a problem which does not require an extensive uses of statistical data. As economic theory advances use of statistical methods also increase. The laws of economics like law of demand, law of supply etc can be considered true and established with the help of statistical methods. Statistics of consumption tells us about the relative strength of the desire of a section of people. Statistics of production describe the wealth of a nation. Exchange statistics through light on commercial development of a nation. Distribution statistics disclose the economic conditions of various classes of people. There for statistical methods are necessary for economics.

Graphs of frequency distribution

A frequency distribution is an overview of all distinct values in some variable and the number of times they occur. A frequency distribution can be represented graphically in any of the following ways. The most commonly used graphs and curves for representation of a frequency distribution are:

a. Histogram

b. Frequency Polygon

c. Smoothened frequency curve

d. Ogives or cumulative frequency curves

e. Pie chart

Histogram: A histogram is a type of graph that has wide applications in statistics. Histograms provide a visual interpretation of numerical data by indicating the number of data points that lie within a range of values. These range of values are called classes or bins. The frequency of the data that falls in each class is depicted by the use of a bar. The higher that the bar is, the greater the frequency of data values in that bin.

· It is a set of vertical bars.

· the variable is plotted on the X axis and the frequencies on the Y axis

· the width of the bars in the histogram will be proportional to the class interval.

· the bars are drawn without leaving space between them.

· If the class intervals are uniform for a frequency distribution, then the width of all the bars will by equal.

Frequency Polygon (or line graphs): Frequency Polygon is a graph of frequency distribution. The purpose of a frequency polygon is to communicate the shape of distributions. It make it easy to compare two or more distributions on the same set of axes. The purpose is similar to that of a histogram but is preferable when it comes to providing data comparisons.

There are two ways of constructing a frequency polygon.

a) Draw histogram of the data and then join by straight lines the mid points of upper horizontal sides of the bars. Join both ends of frequency polygon with x axis. Then we get frequency polygon.

b) Another method of constructing frequency polygon is to take the mid points of the various class intervals and then plot frequency corresponding to each point and to join all these points by a straight line.

Frequency Curves: A continuous frequency distribution can be represented by a smoothed curve known as frequency curve. The mid values of classes are taken along the x axis and the frequencies along y axis. The points thus plotted are joined by smoothened curve. When the points of a frequency polygon are joined by free hand method curve and not by a straight line, we get frequency curve.

Ogives (Cumulative frequency curve): Cumulative frequency is defined as the running total of frequencies. It is the sum of all the previous frequencies up to the current point. It is easily understandable through a Cumulative Frequency Table. Cumulative Frequency is an important tool in Statistics to tabulate data in an organized manner. Whenever you wish to find out the popularity of a certain type of data, or the likelihood that a given event will fall within certain frequency distribution, a cumulative frequency table can be most useful. Say, for example, the Census department has collected data and wants to find out all residents in the city aged below 45. In this given case, a cumulative frequency table will be helpful.

A curve that represents the cumulative frequency distribution of grouped data on a graph is called a Cumulative Frequency Curve or an Ogive. Representing cumulative frequency data on a graph is the most efficient way to understand the data and derive results.

A frequency distribution when cumulated, we get cumulative frequency distribution. A series can be cumulated in two ways. One method is frequencies of all the preceding classes one added to the frequency of the classes. This series is called less than cumulative series. Another method is frequencies of succeeding classes are added to the frequency of a class. This is called more than cumulative series. Smoothed frequency curves drawn for these two cumulative series are called cumulative frequency curve or Ogives. Thus corresponding to the two cumulative series we get two ogive curves, known as less than ogive and more than ogive.

How to plot a More than type Ogive:

· In the graph, put the lower limit on the x-axis

· Mark the cumulative frequency on the y-axis.

· Plot the points (x,y) using lower limits (x) and their corresponding Cumulative frequency (y)

· Join the points by a smooth freehand curve. It looks like an upside down S.

How to plot a Less than type Ogive:

· In the graph, put the upper limit on the x-axis

· Mark the cumulative frequency on the y-axis.

· Plot the points (x,y) using upper limits (x) and their corresponding Cumulative frequency (y)

· Join the points by a smooth freehand curve. It looks like an elongated S.

Pie Diagrams: used when the aggregate and their division are to be shown together. The aggregate is shown by means of a circle and the division by the sectors of the circle. For example: to show the total expenditure of a government distributed over different departments like agriculture, irrigation, industry, transport etc. can be shown in a pie diagram. In constructing a pie diagram the various components are first expressed as a percentage and then the percentage is multiplied by 3.6. so we get angle for each component. Then the circle is divided into sectors such that angles of the components and angles of the sectors are equal. Therefore one sector represents one component. Usually components are with the angles in descending order are shown.

Measures of central tendency

Measures of central tendency—mean, median, and mode—can help you capture, with a single number, what is typical of a data set. They are used to a) To get a single value that describes the characteristics of the entire group; and b) To facilitate comparison.

The mean is the average value of all the data in the set. The median is the middle value in a data set that has been arranged in numerical order so that exactly half the data is above the median and half is below it. The mode is the value that occurs most frequently in the set. In a normal distribution, mean, median and mode are identical in value.

Arithmetic mean

The arithmetic mean is a measure of central tendency in descriptive statistics which shows the average value of a characteristic in a given statistical sample. It is the most popular measure, vitally important, and applied in virtually every area of life. It can be calculated only for quantitative characteristics - such as age, points, height, salary, or weight. Examples include - average daily television viewing time for citizens of a given country, average volume of coffee drunk by a typical American, average annual temperature in your city, or the average amount you spend on food in a typical week.

Arithmetic Mean – Individual observation or individual data series

Below we present a set of measurements containing the number of hours spent by shoppers before making a final purchasing decision in a garment store :

{ 10, 11, 12, 14, 15, 16, 17, 18, 18, 19, 20 }

Note that the number of hours spent varies among the buyers. The measurements are presented individually for each buyer. This is why such a data set is called an individual data series is, therefore, a set of all observations of a given characteristic, usually in ascending order.

a. Add up all the values of the variables X and find out ∑X

b. Divide ∑X by their number of observations N

The mean for the values X1, X2, X3,……….., Xn shall be denoted by clip_image002[1]. Following is the mathematical representation for the formula for the arithmetic mean or simply, the mean.

clip_image004

Arithmetic mean - Discrete series

Discrete series means where frequencies of a variable are given but the variable is without class intervals. When values are frequently repeated in a given set of values for a given characteristic, it is better to use a discrete data series. Each group represents a specific value of the characteristic. It is only possible when the values are discrete, i.e. they come from a countable set: age in years, points in a test.

A discrete variable is a variable whose value is obtained by counting - number of students present, number of red marbles in a jar, number of heads when flipping three coins, students’ grade level …

To find out the total items in discrete series, frequency of each value is multiplies with the respective size. The value so obtained is totaled up. This total is then divided by the total number of frequencies to obtain arithmetic mean.

Steps

1. Multiply each size of the item by its frequency fX

2. Add all fX i.e. (∑f X) f1X1 + f2X2 + ……. + fkXk = ∑fX

3. Divide ∑fX by total frequency (N). f1 + f2 + ……… + fk = ∑f

clip_image026

Arithmetic mean - Continuous series

In continuous frequency distribution, the value of each individual frequency distribution is unknown. A continuous variable is a variable whose value is obtained by measuring - height of students in class, weight of students in class, time it takes to get to school, distance traveled between classes…

Time it takes a computer to complete a task. You might think you can count it, but time is often rounded up to convenient intervals, like seconds or milliseconds. Time is actually a continuum: it could take 1.3 seconds or it could take 1.333333333333333… seconds.

A person’s weight. Someone could weigh 180 pounds, they could weigh 180.10 pounds or they could weigh 180.1110 pounds. The number of possibilities for weight are limitless.

Income. You might think that income is countable (because it’s in dollars) but who is to say someone can’t have an income of a billion dollars a year? Two billion? Fifty nine trillion? And so on…

Age. So, you’re 25 years-old. Are you sure? How about 25 years, 19 days and a millisecond or two? Like time, age can take on an infinite number of possibilities and so it’s a continuous variable.

The price of gas. Sure, it might be $4 a gallon. But one time in recent history it was 99 cents. And give inflation a few years it will be $99. not to mention the gas stations always like to use fractions (i.e. gas is rarely $4.47 a gallon, you’ll see in the small print it’s actually $4.47 9/10ths

Therefore an assumption is made to make them precise or on the assumption that the frequency of the class intervals is concentrated at the center that the mid-point of each class intervals has to be found out. In continuous frequency distribution we can come across sets with many observations, many different values, or values that are unwieldy (such as precise measurements of height or precisely calculated salaries), for which it is difficult to create groups in order to calculate the mean of a discrete data series. In such cases we can use a continuous data series, which is well suited to continuous characteristics (for which measurements are precise and often expressed as decimal numbers). A continuous data series divides all values into groups but somewhat differently than the discrete data series. Each grouping of observations represents not a specific value but a range of values.

1. Find out the mid value of each group or class. The mid value is obtained by adding the lower and upper limit of the class and dividing the total by two. (symbol = m)

2. Multiply the mid value of each class by the frequency of the class. In other words m will be multiplied by f.

3. Add up all the products - ∑fm

4. ∑fm is divided by N

clip_image026

Weighted average:

Weighted average refers to the mathematical practice of adjusting the components of an average to reflect the importance of certain characteristics.

Simple arithmetic mean gives equal importance to all items. Sometimes the items in a series may not have equal importance. So the simple arithmetic mean is not suitable for those series and weighted average will be appropriate.

Weighted means are obtained by taking in to account these weights (or importance). Each value is multiplied by its weight and sum of these products is divided by the total weight to get weighted mean.

When the values are not of equal importance, we assign them certain numerical values to express their relative importance. These numerical values are called weights. If X1, X2, ……, Xk have weights W1, W2, ……., W3, then the weighted arithmetic mean or the weighted mean, which is denoted as clip_image034, is calculated by the following formula;

clip_image036

Median

Median is the value of item that goes to divide the series into equal parts. Median may be defined as the value of that item which divides the series into two equal parts. Arranging the date is necessary to compute median. As distinct from the arithmetic mean, which is calculated from the value of every item in the series, the median is what is called a positional average. The term position refers to the place of value in a series.

The median of set of values arranged either in ascending order or descending order of their magnitude is referred as the middle value. Median is denoted by clip_image002

Calculation of median - Individual observations

Arrange data in ascending or descending order of magnitude.

Where the number of values in a data is odd, the median shall be the middle value. And where the number of values is even, the median shall be the mean of two middle values. Once the distribution is divided into two halve by way of median then the number of values greater than the median is equal to the values smaller than the median.

i. The median of values; 4, 5, 6, 8, 10, 11 and 12 is 8.

ii. The median of values; 4, 6, 7, 9, 11 and 13 is clip_image004

When the number of values is odd, the median is the middle values and when the number of values is even, the median is the mean of the two middle values present in the data. In both the cases the median is the value of clip_image006th item from either ends of the data which is being arranged in ascending or descending order.

Calculation of median - Discrete series

Steps:

1. Arrange the date in ascending or descending order

2. Find cumulative frequencies

3. Apply the formula Median

In an array of values the position, the position of median could be determined by the following formula;

clip_image014

Calculation of median – Continuous series

In case of frequency distribution the median is the value of clip_image016th item from either end. Therefore, if we have 100 items in a frequency distribution, the median will be the value of the 50th item. In order to find the median from a frequency distribution, we need to form a separate column for cumulative frequency. The median will lie in the class which corresponds to the cumulative frequency in which clip_image018lies.

The formula for median in case of frequency distribution is as follows;

clip_image020

Where,

l = lower level of the median class, which is corresponding to the cumulative frequency in which clip_image018[1]lies.

h = class interval size of the median class.

f = frequency of the median class.

n = number of values, or the total frequency.

C.F = cumulative frequency of the class preceding the median class.

Issues in Comparing the mean and median

When house prices are referred to in newspapers, generally the median price is quoted. Why is this measure used instead of the mean?

There are many moderately priced houses, but there are also some expensive ones and a few very expensive ones. The mean figure could be quite high as it includes the prices of the more expensive houses. But the median gives a more accurate and realistic value of the prices faced by most people.

In summary, the median is the central number and is good to use in skewed (or unbalanced) distributions because it is not affected by outliers.

Suppose you want to know how much money a family could afford to spend on housing. This would depend on the total family income.

For a family of five (two parents who work and three children with no income) the mean income of each family member is the total income divided by five (e.g., 60,000 ÷ 5 = 12,000). However, the median income would be zero, because more than half of the members of the family make nothing. In some situations, the mean can be much more informative than the median.

If you want to find out whether a country is wealthy or not, you might consider using the median as your measure of central tendency instead of the mean.

The mean family income could be quite high if income is highly concentrated in a few very wealthy families (despite the fact that most families might earn essentially nothing). Thus, the median family income would be a more meaningful measure—at least half the families would earn the median income or less, and at least half would earn at least as much as the median income or more.

Suppose you are applying for a job as an accountant at several large firms, and you want to get an idea of how much money you could expect to be making in five years if you join a particular firm. You may want to consider the salaries of accountants in each firm five years after they are hired.

One very high salary could make the mean salary higher; that might not reflect a typical salary within these firms. However, half the accountants make the median salary or less, and half make the median salary or more. So, the measure of central tendency that would give you a better idea of a typical salary would be the median.

By choosing a measure of central tendency favourable to your point of view, you can mislead people with statistics. In fact, this is commonly done.

Imagine you are the owner of a bakery that makes and sells individual birthday cakes and huge wedding cakes.

It might be in your interest to claim to your customers that the prices have been lowered, and to claim to your shareholders that you have raised the prices. Suppose that last year you sold 100,000 birthday cakes at $10 each, and 1,000 wedding cakes at $1,000 each. This year, you sold 100,000 birthday cakes at $8 each and 1,000 wedding cakes at $1,200 each.

The median price of the 101,000 cakes sold last year is $10, because more than half of the items sold were birthday cakes. The median price of the 101,000 cakes sold this year is $8.

The mean price of the 101,000 cakes sold last year is $19.80. (100,000 x $10 + 1,000 x $1,000) ÷ 101,000 = $19.80

The mean price of the 101,000 cakes sold this year is also $19.80. (100,000 x $8 + 1,000 x $1,200) ÷ 101,000 = $19.80

The average price per cake sold is the same in both years. Also, the total revenue and the number of the cakes sold was the same. The idea is that you can make data appear to tell conflicting stories by choosing the appropriate measure of central tendency.

It is important to note that you do not have to use only one measure of central tendency. The mean and median can both be used, thus providing more information about the data.

Mode

The mode is the most frequent score in our data set. On a histogram it represents the highest bar in a bar chart or histogram. You can, therefore, sometimes consider the mode as being the most popular option. An example of a mode is presented below:

Mode is the common item of a series. It is the value which occurs the greatest number of frequency in a series. It is derived from the French word “La mode” meaning fashion. Mode is the most fashionable or typical value of a distribution, because it is repeated the highest number of times in the series.

Mode - individual observations

Mode can be often be found out by mere inspection in case of individual observations. The data have to be arranged in the form of array so that the value which has the highest frequency can be known for example to persons has the following income.

850, 750, 600, 825, 850, 725, 600, 850, 640, 530

Here 850 repeated three times; therefore the mode salary is 850.

In certain cases that there may not be a mode or there may be more than one mode.

For example:

a) 40,44 45,48,52 (No mode)_

b) 45, 55, 25, 23, 28, 32, 55, 45 (bi modal mode is i) 45

Calculation of mode: discrete series

In discrete series the value having highest frequency is taken as Mode. A glance at a series can reveal which is the highest frequency. So we get mode by mere inspection. So this method is also called inspection method.

Example:

Find the mode from following data

Size : 5 15 16 25 37 45 56

Frequency: 10 16 28 15 30 40 38

Ans: the value 45 has highest frequency

Calculation of Mode: Continuous Series

Step1: By preparing grouping table and analysis table or by inspection ascertain in the modal class.

Step 2: Determine the value of mode by applying the formula

L = Lower limit of the modal class,

F1 = frequency of the model class;

F2 = frequency of the class preceding the modal class

F3 = frequency of the class succeeding the modal class

I = the size of the class interval of the modal class.

Circumstances generally dictate which measure of central tendency—mean, median or mode—is the most appropriate. If you are interested in a total, the mean tends to be the most meaningful measure of central tendency because it is the total divided by the number of data. For example, the mean income of the individuals in a family tells you how much each family member can spend on life's necessities. The median measure is good for finding the central value and the mode is used to describe the most typical case.

Positional Values

Quartiles, deciles and percentiles Median divides the distribution in to two equal parts. There are other values also which divide the series in to a number of equal parts and they are called partition values or positional value. There are 4 types of positional values. They are median, quartiles, deciles and percentile.

1. Quartiles

The values which divide an array (a set of data arranged in ascending or descending order) into four equal parts are called Quartiles. The first, second and third quartiles are denoted by Q1, Q2,Q3 respectively. The first and third quartiles are also called the lower and upper quartiles respectively. The second quartile represents the median, the middle value. The first quartile (Q1) or lower quartile, has 25% of the items of the distribution below it and 75% of the items are greater than it. Q2 (median) has 50% of the observations above it and 50% of the observations below it. The upper quartile (Q3), has 75% of the items of the distribution below it and 25% of the items are above it.

Quartiles for Individual observation and discrete series :

clip_image004

clip_image006

clip_image008

For Example:

Following is the data of marks obtained by 20 students in a test of statistics;

53

74

82

42

39

20

81

68

58

28

67

54

93

70

30

55

36

37

29

61

In order to apply formulae, we need to arrange the above data into ascending order i.e. in the form of an array.

20

28

29

30

36

37

39

42

53

54

55

58

61

67

68

70

74

81

82

93

Here, n = 20

i. clip_image010

clip_image012

clip_image014

The value of the 5th item is 36 and that of the 6th item is 37. Thus, the first quartile is a value 0.25th of the way between 36 and 37, which are 36.25. Therefore, clip_image016= 36.25. Similarly,

clip_image018

clip_image020

clip_image022

The value of the 10th item is 54 and that of the 11th item is 55. Thus the second quartile is the 0.5th of the value 54 and 55. Since the difference between 54 and 55 is of 1, therefore 54 + 1(0.5) = 54.5. Hence, clip_image024= 54.5. Likewise,

clip_image026

clip_image028

clip_image030

The value of the 15th item is 68 and that of the 16th item is 70. Thus the third quartile is a value 0.75th of the way between 68 and 70. As the difference between 68 and 70 is 2, so the third quartile will be 68 + 2(0.75) = 69.5. Therefore, clip_image032= 69.5.

Quartiles for continuous series:

The quartiles may be determined from grouped data in the same way as the median except that in place of n/2 we will use n/4. For calculating quartiles from grouped data we will form cumulative frequency column. Quartiles for grouped data will be calculated from the following formulae;

clip_image034

clip_image036

clip_image024[1]= Median.

Where,

l = lower limit of the class containing the clip_image038, i.e. the class corresponding to the cumulative frequency in which n/4 or 3n/4 lies

h = class interval size of the class containingclip_image040.

f = frequency of the class containing clip_image038[1].

n = number of values, or the total frequency.

C.F = cumulative frequency of the class preceding the class containing clip_image038[2].

2. Deciles:

The values which divide an array into ten equal parts are called deciles. The first, second,…… ninth deciles by clip_image064respectively. The fifth decile (clip_image066 corresponds to median. The second, fourth, sixth and eighth deciles which collectively divide the data into five equal parts are called quintiles.

Deciles for individual and discrete series

clip_image068

clip_image070

clip_image076

For Example:

We will calculate second, third and seventh deciles from the following array of data.

20

28

29

30

36

37

39

42

53

54

55

58

61

67

68

70

74

81

82

93

i. clip_image078

clip_image080

clip_image082

The value of the 4th item is 30 and that of the 5th item is 36. Thus the second decile is a value 0.2th of the way between 30 and 36. The fifth decile will be 30 + 6(0.2) = 31.2. Therefore, clip_image084= 31.2.

ii. clip_image086

clip_image088

clip_image090

The value of the 6th item is 37 and that of the 7th item is 39. Thus the third decile is 0.3th of the way between 37 and 39. The third decile will be 37 + 2(0.3) = 37.6. Hence, clip_image092= 37.6.

iii. clip_image094

clip_image096

clip_image098

The value of the 14th item is 67 and that of the 15th item is 68. Thus the 7th decile is 0.7th of the way between 67 and 68, which will be as 37 + 0.7 = 67.7. Therefore, clip_image100= 67.7.

Decile for Continuous series

Decile for continuous series can be calculated from the following formulae;

clip_image102

clip_image104

clip_image106

Where,

l = lower limit of the class containing the clip_image108, i.e. the class corresponding to the cumulative frequency in which 2n/10 or 9n/10 lies

h = class interval size of the class containingclip_image108[1].

f = frequency of the class containing clip_image108[2].

n = number of values, or the total frequency.

C.F = cumulative frequency of the class preceding the class containingclip_image108[3].

3. Percentile

The values which divide an array into one hundred equal parts are called percentiles. The first, second, … Ninety-ninth percentile are denoted by clip_image130 The 50th percentile (clip_image132) corresponds to the median. The 25th percentile clip_image134corresponds to the first quartile and the 75th percentile clip_image136corresponds to the third quartile.

Percentiles for individual and discrete series:

Percentile could be calculated from the following formulae;

clip_image138

clip_image140 clip_image074[3]

clip_image142

For Example:

We will calculate fifteenth, thirty-seventh and sixty-fourth percentile from the following array;

20

28

29

30

36

37

39

42

53

54

55

58

61

67

68

70

74

81

82

93

i. clip_image144

clip_image146

clip_image148

The value of the 3rd item is 29 and that of the 4th item is 30. Thus the 15th percentile is 0.15th item the way between 29 and 30, which will be calculated as 29 + 0.15 = 29.15. Hence, clip_image150= 29.15.

ii. clip_image152

clip_image154

clip_image156

The value of 7th item is 39 and that of the 8th item is 42. Thus the 37th percentile is 0.77th of the between 39 and 42, which will be calculate as 39 + 3(0.77) = 41.31. Hence, clip_image158= 41.31.

iii. clip_image160

clip_image162

clip_image164

The value of the 13th item is 61 and that of the 14th item is 67. Thus, the 64th percentile is 0.44th of the way between 61 and 67. Since the difference between 61 and 67 is 6 so 64th percentile will be calculated as 61 + 6(0.44) = 63.64. Hence, clip_image166= 63.64.

Percentiles for Continuous series:

Percentiles can also be calculated for grouped data which is done with the help of following formulae;

clip_image168

clip_image170

clip_image172

Where,

l = lower limit of the class containing the clip_image174, i.e. the class corresponding to the cumulative frequency in which 35n/100 or 99n/100 lies

h = class interval size of the class containing.clip_image176.

f = frequency of the class containingclip_image174[1].

n = number of values, or the total frequency.

C.F = cumulative frequency of the class preceding the class containingclip_image174[2].