Grouped data is data that has been grouped together in categories. Mean of a set of observations is the average of the given observations. To find the mean, we divide the sum of the observations by the total number of observations. However, for grouped data, we cannot find the exact Mean, we can only give estimates.
In statistics, Mean, Median and Mode are known as the measures of central tendency.
The mean of a given set of data is equal to the sum of the numerical values of each and every observation divided by the total number of observations.
i..e, for n observations x1, x2, x3, … , xn, the arithmetic mean is:
span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="x=i=1nxin">x¯¯¯=∑i=1nxinx¯=∑i=1nxin
Mean of grouped data
There are three methods for finding the mean from a grouped frequency table.
Method 1: Direct method
Mid-point or Class mark (xi) = spanid="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="UpperclasslimitLowerclasslimit2">Upper class limit − Lower class limit2Upper class limit − Lower class limit2
Then the mean span id="MathJx-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="x=i=1nfixii=1nfi">x¯¯¯=∑i=1nfixi∑i=1nfix¯=∑i=1nfixi∑i=1nfi
where fi is the frequency of a class interval
Method 2: Assumed Mean method
If 'a' represents the assumed mean of a given frequency distribution,
then the deviation of the assume mean from each class mark is di = xi – a
and the mean is span id="MathJaxElement-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="x=a+i=1nfidii=1nfi">x¯¯¯=a+∑i=1nfidi∑i=1nfix¯=a+∑i=1nfidi∑i=1nfi
Method 3: Step Deviation method
The step deviation method is suitable if the deviations of the class marks from the assumed mean are large and they all have a common factor.
Mean span d="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="x=a+hi=1nfiuii=1nfi">x¯¯¯=a+h∑i=1nfiui∑i=1nfix¯=a+h∑i=1nfiui∑i=1nfi
where span i="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="ui=(xia)h">ui=(xi–a)hui=(xi–a)h
'a' is the assumed mean
fi is the frequency of a class interval
h is the class size
Mode of a given set of numbers is a the ‘most popular’ number. The word comes from the French expression ‘a la mode’ which means ‘fashionable’. There are many cases in which the arithmetic mean and median do not give us the true characteristics of a given set of data. For these, the mode indicates the most preferred value or in other words, the observation with highest frequency.
For grouped data, we cannot find the exact value of mode. In this case, we find the modal class. This is particularly useful in cases where the values are not numerical but qualitative, for example the preference of consumers across various brands of a particular product.
The first step in finding the mode of a grouped data is to locate the class interval with the maximum frequency. The class interval corresponding to the maximum frequency is called the modal class.
The mode of this data is then calculated as:
Mode = span id="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1+">1+1+span id="MathJax-Element-8-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="f1f02f1f0f2">f1−f02f1−f0−f2f1−f02f1−f0−f2 span id="MathJax-Element-9-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="h">×h×h
- l = lower limit
- h = size of the class interval
- span id="MathJax-Element-10-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="f1">f1f1 = frequency of the modal class
- span id="MathJax-Element-11-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="f0">f0f0 = frequency of the class preceding the modal class
- span id="MathJax-Element-12-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="f2">f2f2 = frequency of the class succeeding the modal class
Median is the middle most value of the observations when the observations are either arranged in increasing or decreasing order. The median is less sensitive to extreme values and hence is preferred over mean in cases where the range of the data is large.
It is easy to find the median when it is a discrete data. For grouped data, however, we cannot find an exact value for the median, but we can find the class which contains the median.
Median is the middle most value of the observations when the observations are arranged either in the increasing or decreasing order.
Let n be the total number of observations in a given data.
If n is odd, then the median is the value of span id="MathJax-Element-13-Frame" class="MatJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="n+12">n+12n+12th observation.
If n is even, then the median is the average or arithmetic mean of the values of the span id="MathJax-Element-14-Frame" class="MathJax" style="box-sizing: border-box; marin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="n2">n2n2 th and span id="MthJax-Element-15-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="n+12">n+12n+12th observations.
Calculating the median for grouped data:
Preparing a cumulative frequency distribution table is the first step in calculating the median for grouped data. Next we need to identify what is known as the median class.
The cumulative frequency of a class is obtained by adding the frequencies of all the classes preceding the given class.
Median for grouped data = span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="l+[n2cff]h">l+[n2−cff]hl+[n2−cff]h
Where l is lower class limit of the median class
n is the total number of observations.
cf is the cumulative frequency of the class preceding the median class.
f is the frequency of the median class and h is the class size.
4. Graphical Representation of Cumulative Frequency Distribution
In ballistics or aerodynamics, an ogive is a pointed, curved surface mainly used to form the approximately streamlined nose of a bullet or other projectile. In Gothic architecture, ogives are the intersecting ribs of arches that define the surface of a Gothic vault. An ogive or ogival arch is a pointed, "Gothic" arch, drawn with compasses. In woodworking, an ogive is a type of curve a piece of wood can be shaped in.
In statistics, an ogive is a free-hand graph showing the curve of a cumulative distribution function. The points plotted are the upper class limit and the corresponding cumulative frequency. Defined more formally, a curve that represents the cumulative frequency distribution of grouped data is called an ogive or cumulative frequency curve.
Ogives or cumulative frequency curves
A curve that represents the cumulative frequency distribution of a grouped data is called an Ogive or cumulative frequency curve.
There are two types of Ogives, the more than type Ogive and the less than type Ogive.
An Ogive curve representing a cumulative frequency distribution of ‘more than’ type is called a more than Ogive.
An Ogive curve representing a cumulative frequency distribution of ‘less than’ type is called a less than Ogive.
Finding median using Ogive curves
Ogives can be used to find the median of grouped data.
The median can be obtained graphically by plotting the Ogives of less than type and more than type and locating the point of intersection of both the Ogives. The x-coordinate of the point of intersection of the two Ogives gives the median of the grouped data.
Note: The median can be found graphically even if only one Ogive is given, by following the steps below:
- locate the mid-point (corresponding to n/2) on the Y-axis,
- draw a line parallel to the x-axis from this point,
- drop a perpendicular to the x-axis from the point where the line meets the Ogive
The point where the perpendicular meets the x-axis is the median.