Monday, April 6, 2015

Measures of Variation in Excel

Measures of Variation in

Excel

Measures of Variation Overview

Measures of variation describe the degree of spread in a random variable. Measures of variation are sometimes referred to as measures of dispersion. Measures of variation are properties describing probability distributions, populations, or samples taken from a population. Quite often the same measures of variation are calculated differently for populations and samples taken from those population.

The following measures of variation can be calculated in Excel and will be discussed here:

σ2 = Population variance – A measure of the degree of dispersion of values of all of the points contained within a population. σ2 equals the sum of the squared differences between the sample mean and each data value divided by N. Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units.

s2 = Sample variance – A measure of the degree of dispersion of values of all of the points contained within a sample. s2 equals the sum of the squared differences between the sample mean and each data value divided by (n – 1).

σ = Population standard deviation – The square root of population variance. Standard deviations are not additive but are expressed in the same units as the data values being described.

s = Sample standard deviation – The square root of sample variance.

SS = Sum of squares - Because variances are additive, the sum of the squares of differences between a mean and data values is used to describe the total variance within specific categories. ANOVA and linear regression both uses the sum of the squares in their calculations.

MAD – Mean absolute deviation and Median absolute value – Mean absolute deviation equals the average distance (absolute distance) between data values and the mean. Median absolute value, which is the average distance between data values and the median, is more robust (less sensitive to outliers) than mean absolute deviation.

R = Range – Range describes the difference between the largest and smallest values of a data set. Range is sensitive to outliers. Range is used in SPC (Statistical Process Control) to describe the spread within each sample when sample size is small (typically 2 – 9). Standard deviation is used in SPC to describe spread within each sample when sample size become larger.

Interquartile range – Interquartile range is the spread or width of the middle two quartiles of the four quartiles that all data of a data set have been divided into. Quartiles divide rank-ordered data into four equal parts, i.e., quartiles Q1, Q2, Q3, and Q4 which contains the highest values. The box of a boxplot contains the middle two quartiles of data while the whiskers contain the outer quartiles of data. Interquartile range is very helpful in finding outliers.

Excel also has excellent tools to visually display the degree of variation existing in a data set. The Excel tools that will be described in this blog article that visually display variation of data are the following:

  1. Histogram

  2. Box plot

  3. Scatter plot

Population Variance

Population variance is a measure of the degree of dispersion of values of all of the points contained within a population. σ2 (sigma squared) equals the sum of the squared differences between the sample mean and each data value divided by N, the total number of data values in a population. µ (mu) is the population mean. The formula for calculating population variance is the following:

Variation in Excel - Population Variance
(Click On Image To See a Larger Version)

Population variance is calculated in Excel 2010 and later with the following formula:

σ2 = VAR.P(data range)

Excel 2007 and earlier used the following formula, which also works in later Excel versions:

σ2 = VARP(data range)

Note that Greek letters are often used to denote population parameters while letters from the English alphabet are often used to denote sample statistics.

Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units. The square root of the variance is the standard deviation. Standard deviations are not additive but they are conveniently expressed in the same units as the original data.

It is important to note that population variance is calculated by dividing the sum of the squared by N, the total number of data points in the population. Sample variance, on the other hand, is calculated by dividing the sum of the squares by (n – 1) instead of n, the number of data points in the sample. This will be discussed in detail shortly.

 

Sample Variance

Sample variance is a measure of the degree of dispersion of values of all of the points contained within a sample. s2 equals the sum of the squared differences between the sample mean and each data value divided by (n-1), the total number of data values in sample minus 1. Xavg is the sample mean. The formula for calculating population variance is the following:

Variation in Excel - Sample Standard Deviation
(Click On Image To See a Larger Version)

Sample variance is calculated in Excel 2010 and later with the following formula:

s2 = VAR.S(data range)

Excel 2007 and earlier used the following formula, which also works in later Excel versions:

s2 = VAR(data range)

 

Unbiased Sample Variance

Dividing the sum of the squares from sample data by n instead of (n-1) would produce a biased estimate of the sample variation. The term n-1, known as Bessel’s correction, replaces n as the divisor to remove the bias.

An intuitive explanation for the use of Bessel’s correction when calculating sample variance or sample standard deviation is as follows:

It is usually the case that when estimating the population standard deviation, σ, from a sample standard deviation, s, the population mean, µ, is also unknown. The sample standard deviation is therefore calculated using the sample mean, xavg, in place of the population mean, µ. This substitution creates a slight bias because the sample’s data points are likely to be closer to xavg than to µ. The sum of the squares calculated from (xi – xavg)2 will be smaller than (xi - µ)2. This is compensated for by using Bessel’s correction (n-1) as the divisor instead of n when calculating sample standard deviation or sample variance.

Note that if the population mean, µ, is known and used instead of sample mean, xavg, during the calculation of sample standard deviation or sample variance, then Bessel’s correction should not be used; the divisor should be n instead of n-1. This is usually not the case however.

Bessel’s correction is a consistent estimator because it converges in probability to the population value as the sample size goes to infinity. As sample size increases, the need for the use of Bessel’s correction decreases. When n exceeds 75, the difference between the biased sample standard deviation (using n as the divisor instead of n-1) and the population standard deviation is generally less than 1 percent. The use of uncorrected sample standard deviation for very large samples is generally acceptable.

 

Derivation of Bessel’s Correction

Bessel’s Correction can be derived in a number of ways. Here is one method that is straightforward and intuitive.

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

Variation in Excel - Bessel's Correction Derivation
(Click On Image To See a Larger Version)

 

Population Standard Deviation

Population standard deviation, σ (sigma), is the square root of population variance, σ2, and is calculated using the following formula:

Variation in Excel - Population Standard Deviation

µ (mu) = Population mean

N = total number of data points in the population

Population standard deviation is calculated in Excel 2010 and later with the following formula:

σ = STDEV.P(data range)

Excel 2007 and earlier used the following formula, which also works in later Excel versions:

σ = STDEVP(data range)

Note that Greek letters are often used to denote population parameters while letters from the English alphabet are often used to denote sample statistics.

Variances are additive but are not expressed in the same measure as the data values being described. The square root must be taken to get back to the original units. The square root of the variance is the standard deviation. Standard deviations are not additive but they are conveniently expressed in the same units as the original data.

It is important to note that population standard deviation is calculated by dividing the sum of the squared by N, the total number of data points in the population. Sample standard deviation, on the other hand, is calculated by dividing the sum of the squares by (n – 1) instead of n, the number of data points in the sample.

 

Sample Standard Deviation

Sample standard deviation, s, is the square root of sample variance, s2, and is calculated using the following formula:

Variation in Excel - Sample Standard Deviation
(Click On Image To See a Larger Version)

xavg = Population mean

n = total number of data points in the sample

Sample standard deviation is calculated in Excel 2010 and later with the following formula:

s = STDEV.S(data range)

Excel 2007 and earlier used the following formula, which also works in later Excel versions:

s = STDEV(data range)

 

Descriptive Statistics in Excel

Excel provides a tool in the Data Analysis toolpak called Descriptive Statistics that calculates a data column’s sample standard deviation and sample variance along with sample statistics. The Data Analysis toolpak is an Excel add-in that is available in most versions of Excel. This add-in is initially inactive and must be activated by the user before the first use. After this add-in has been activated, Descriptive Statistics can be accessed as follows:

Data tab / Data Analysis / Descriptive Statistics

The Descriptive Statistics dialogue box then appears. The data to which Descriptive Statistics will be applied needs to be arranged in a single column. An example of a column of sample data and the completed Descriptive Statistics dialogue box is shown as follows:

Variation in Excel - Descriptive Statistics - Variance
(Click On Image To See a Larger Version)

Clicking OK will produce the following output. The Excel formulas needed to calculate the same results are shown as well.

Variation in Excel - Descriptive Statistics - Variance
(Click On Image To See a Larger Version)

Note that the calculations of standard deviation and variance are done for samples using Bessel’s correction n-1 as the divisor of the sum of squares and not n. Descriptive Statistics is intended to provide sample statistics and not population parameters.

Sum of Squares

SS = Sum of squares - Because variances are additive, the sum of the squares of differences between a mean and data values is used to describe the total variance within specific categories. The Excel formula to calculate the sum of the squares of deviations of sample points from the sample mean is as follows:

DEVSQ(data range)

This formula is the same for all versions of Excel.

SS = DEVSQ(data range) = ∑ (xi – xavg)

ANOVA uses the sum of the squares to calculate the within-sample variance, SSwithin-groups, and the between-sample variance, SSbetween-groups. The F Value for each F test in ANOVA is calculated as follows:

F Value = [ SSbetween-groups / dfbetween-groups ] / [SSwithin-groups / dfwithin-groups ]

The p Value for each F test in ANOVA is calculated in Excel as follows:

p Value = F.DIST.RT( F Value, dfbetween-groups , dfwithin-groups )

If the calculated p Value is smaller than the specified alpha, the factor that is being evaluated in the F test is deemed to be significant, i.e., has an effect on the data values.

Linear regression uses the sum of the squares to calculate total variation (SST), explained variation (SSR), and unexplained variation (SSE). R Square of a linear regression is calculated as follows:

R Square = (Explained variation) / (Total variation) = SSR / SST

 

MAD

MAD can refer to either Mean Absolute Deviation (sometimes called Average Absolute Deviation) or Median Absolute Deviation. It is necessary to clarify which of the two is being calculated when deriving MAD for a set of data.

 

Mean Absolute Deviation

Mean Absolute Deviation is calculated by the following formula:

Mean Absolute Deviation = 1/n * ∑ (xi – xavg)

Mean Absolute Deviation can be calculated in all versions of Excel with the following Excel formula:

AVEDEV(data range)

Following is an example of the calculation of Mean Absolute Deviation using both the formula and a more detailed showing each step of the calculation:

Variation in Excel - Mean Absolute Deviation

 

Median Absolute Deviation

Median Absolute Deviation relies on the median and is therefore more robust than Mean Absolute Deviation. Outliers have a much smaller effect on the median than on the mean.

Median Absolute Deviation is calculated by the following formula:

Mean Absolute Deviation = 1/n * ∑ (xi – Median)

Excel does not have a single formula to calculate Median Absolute Deviation as there is for Mean Absolute Deviation. Each individual step of the calculation must be performed as follows:

Variation in Excel - Median Absolute Deviation

Standard Deviation of a Continuous Distribution

This article has so far discussed the calculation of variation of discrete sets of data. The standard deviation of a population of data distributed according to probability function p(x) can be stated in general terms as follows:

Variation in Excel - Standard Deviation of a Continuous Variable
(Click On Image To See a Larger Version)

Many of the individual statistical distributions use specific, unique formulas to calculate standard deviation. For example, the variance of data that are distributed according to the binomial distribution is calculated by the formula np(1-p).

 

Effect of Changing Units

Unit can be changed by either adding a constant to each data measurement or by multiplying each data measurement by a constant. These two methods of altering the units of measuring have different effects on the measured variation for the data set as follows:

Adding a Constant to Each Data Value

Adding a constant to each data value does not change the distance between data values. All measures of variability therefore remain unchanged.

Multiplying Each Data Value By a Constant

Multiplying each data value by a constant multiplies range and standard deviation by the value of the constant. The variance is multiplied by the square of the constant.

 

Range

Range of a data set simply equals the highest data value minus the lowest data value. Excel does not have a formula that specifically calculates the range of a data set. This must be implemented in Excel as follows:

Range = MAX(data range) – MIN(data range)

Range is sensitive only to changes in the outermost data values but not changes to any other data values.

 

Interquartile Range

Quartiles divide rank-ordered data into four equal parts, i.e., quartiles Q1, Q2, Q3, and Q4 which contains the highest values. Each quartile represent a range into which one quarter of the data points fall. The upper and lower endpoints of each quartile are not data points of the data set but are the borders of the interval that contains one quarter of the data points.

The interquartile range is the width of the middle two quartiles combined. This is the distance between the upper boundary of Q2 and the lower boundary of Q3.

The boundaries between the quartiles are calculated in Excel using the QUARTILE.EXC() as shown in the following image. The highest and lowest data points within each quartile are calculated in Excel using MIN(), MAX(), INDEX, and MATCH() as shown in the following:

Variation in Excel - Quartiles
(Click On Image To See a Larger Version)

Here is are close-up images of these Excel commands.

Variation in Excel - Quartiles Close-up
(Click On Image To See a Larger Version)

Variation in Excel - Quartiles Close-up

The Interquartile range is calculated in Excel 2010 and later with the following:

Interquartile range = QUARTILE.EXC(data range,3) – QUARTILE.EXC(data range,1)

In Excel 2007 and prior, the Interquartile range is calculated using QUARTILE(). In Excel 2010 QUARTILE() is replaced by QUARTILE.INC(). This produces a slightly different result than the other Excel 2010 function QUARTILE.EXC().

In Excel 2007 the Interquartile range is calculated as follows:

Interquartile range = QUARTILE(data range,3) – QUARTILE(data range,1)

It should be noted that the median of a data set represent the boundary between the 2nd and 3rd quartiles.

The Excel algorithms for the 3 quartile functions – QUARTILE(), QUARTILE.INC(), and QUARTILE.EXC() – are shown as follows:

QUARTILE(), QUARTILE.INC() algorithms

QUARTILE.EXC() algorithm

 

Visual Presentation of Variation in Excel

Two tools in Excel that provide a visual representation of the spread of one-dimensional data set are the histogram and the box plot. The spread of two-dimensional data (X-Y data) can be visually presented using a scatterplot graph in Excel.

 

Histogram in Excel

A histogram is a bar chart with each bar showing the frequency of occurrence of data values within a specified range. The ranges are called bins. Creating a histogram in Excel requires specifying the upper limits of each bin as follows:

Variation in Excel - Histogram Raw Data

The Excel histogram is one of the tools of the Data Analysis toolpak. The Analysis toolpak is an Excel add-in that is initially inactive in Excel. The user must activate this add-in prior to its first use. The Excel histogram tool can be accessed as follows:

Data tab / Data Analysis / Histogram

Doing so brings up the Histogram dialogue box which should be completed as follows:

Variation in Excel - Histogram Dialogue Box
(Click On Image To See a Larger Version)

Clicking OK produces the following Excel histogram.

Variation in Excel - Histogram
(Click On Image To See a Larger Version)

The following two articles in this blog provide detailed descriptions on the following histogram topics:

  1. Creating a histogram in Excel using the histogram tool:

http://blog.excelmasterseries.com/2014/05/how-to- create-histogram-in-excel-2010_27.html

  1. Creating an automatically updating histogram in 7 steps in Excel using formulas and a bar chart:

http://blog.excelmasterseries.com/2014/05/how-to-create- histogram-in-excel-2010.html
(Click On Image To See a Larger Version)

 

Frequency Tables

The Excel Histogram tool produces a frequency table that is part of its output. The frequency table displays a count of the number of data points that fall into each bin. This is the basis of the histogram chart shown as follows:

Variation in Excel - Histogram

A stand-alone frequency table can also be produced in Excel using the FREQUENCY() formula. The FREQUENCY() formula is an array formula that is implemented as follows.

 

Step 1 – Highlight the Cells That Will Contain the Output

The raw data should be sorted in a column and the upper limits of each bin should also be listed in a column as shown in the following image.

The FREQUENCY() formula produces a frequency table in a separate data column that contains 1 more cell than the number of bins. The upper limits of six bins are listed in this example so there are six total bins. The FREQUENCY() formula will therefore require a column of seven cells to hold its output.

The FREQUENCY() is an array formula. The first step in implementing an array formula in Excel is to highlight (select) the cells that will hold the output. Click and drag the mouse to highlight seven empty cells in a column where the output should go.

Variance_Frequency_1_600

 

Step 2 – Type in the Formula

While the seven cells remain highlighted, type in the FREQUENCY() formula as shown.

Variation in Excel - Frequency
(Click On Image To See a Larger Version)

 

Step 3 – Type CTRL – SHIFT – ENTER at the Same Time

Doing so will produce the frequency table in cells H4:H10 that have been highlighted.

Variation in Excel - Frequency
(Click On Image To See a Larger Version)

 

Box Plots

Box plots visually display the spread of data in each of the four quartiles. The interquartile range, Q2 and Q3, are contained the two boxes. The outer quartiles, Q1 and Q4, are displayed as whiskers above and below the boxes. Q4 is displayed as the upper whiskers and Q1 is displayed as the lower whisker. For this reason boxplots are sometimes called Box-and-Whisker plots.

The boundary between the upper box (Q3) and the lower box (Q2) is the median. Box plot diagrams can also be constructed in Excel to display the data’s mean as well. An example of a box plot in Excel and the data from which it was derived is shown as follows:

Variation in Excel - Box Plot
(Click On Image To See a Larger Version)

The following article in this blog provides step-by-step instructions on how to produce the preceding box plot in 8 steps in Excel:

http://blog.excelmasterseries.com/2015/02/box-plots-in-8-steps- in-excel.html

 

Scatter Plot in Excel

A scatter plot is a diagram using Cartesian coordinates to display the X-Y values of a two-dimensional data set. The Y variable displayed on the vertical axis is often the dependent or response variable while the X variable displayed on the horizontal axis is related to that response variable. The scatter plot is also referred to also referred to as a scatter graph, scatter diagram, or scatter chart.

The following Excel scatter plot displays data points which contain a dependent variable (number of parts produced) as the X value and the residual associated with each dependent variable as the Y value. This scatter plot indicates that the residuals have slightly increasing variance as the dependent variable increases in value. This is indicated by the slight fanning-out of the Y values (the residual values) as the X variable (number of parts produced) increases. This scatter plot was used as part of a single-variable linear regression to determine whether one of the required conditions of linear regression, namely that the residual have constant variance, is met. The slight increase in variance of residuals as the dependent variable increases is not significant enough to invalidate the regression that was performed.

Variation in Excel - Scatter Plot
(Click On Image To See a Larger Version)

 

Variation in Statistical Process Control

Every measurable process contains variation. This variation can be classified in the following two categories:

Controlled Variation – This type of variation is consistent and stable over time and is the result of random statistical variation that is always present at least to some degree.

Uncontrolled Variation – This type of variation changes over time and is attributed to assignable causes. It is the job of management to determine the assignable causes of uncontrolled variation and reduce them as much as possible.

Variation in a measurable process can be displayed on control charts. Control charts in SPC (statistical process control) provide information about samples taken from a process. Individual samples are called subgroups. A subgroup is specific number of measurements periodically taken at a single point in a process.

An SPC control chart plots the location and dispersion of each subgroup. An SPC control chart consists of a pair of graphs; an upper graph charts the location of each successive subgroup and a lower graph charts the dispersion of each successive subgroup. Two common SPC control charts are the following:

X-Bar / R Charts are running records of subgroup average value (X-Bar) and subgroup range (R). Subgroups usually consist of 3 to 5 data measurements.

X-Bar / S Charts are running records of subgroup average value (X-Bar) and subgroup standard deviation (S). Subgroups usually consist of 8 to 10 data measurements.

A process can be deemed to be OOC (out of control) if the location or dispersion values of subgroups fall outside of defined upper or lower control limits or show patterns. The upper and lower control limits are approximately three sigma from the mean value being measured. One sigma is equal to the length of one standard deviation.

The interval between subgroups should be set so that changes between subgroups will be maximized. Subgroups should be taken over an interval that is long enough for potential variation to show up. The sampling interval between individual samples within the subgroups should be set so that variation within subgroups is minimized. Subgroup size should be held constant. Typically at least 20 subgroups are needed to judge statistical control and provide a reasonable estimate of parameters of mean and variation.

 

Excel Master Series Blog Directory

 

Click Here To See a List Of All

Statistical Topics And Articles In

This Blog

 

You Will Become an Excel Statistical Master!

75 comments:

  1. Wow! i can imagine the dedication you gave in order to make this post. Thanks for sharing this. If you'd like i want to share this on Facebook coz this is very informative. I was just scrolling across the web looking for essay writer online and i found your amazing post. Thanks again.

    ReplyDelete
  2. Excel can be a very useful tool that you can use in different reports on a daily basis. As long as you know how to navigate it properly, no wonder students via American-Writers.org reviews are very keen on using this tool.

    ReplyDelete
  3. When you google “paraphrase for me”, you get surprisingly few trustworthy companies. Some offer a limited range of services, while others openly use plagiarism rewriter software instead of professional writers. Here you get top-notch our plagiarism rewriter with your essays, research and term papers. The writers easily deal with all types of assignments. Even a dissertation can be professionally rewritten without a trace of its original structure and language.

    ReplyDelete
  4. Population variance was a complicated topic for me during college education. This is a measure of the degree of dispersion of values of all of the points contained within a population. After reading writing service I found out that it is important to note that population variance is calculated by dividing the sum of the squared by N, the total number of data points in the population.

    ReplyDelete
  5. Awesome interior design that is write my essay because here is some thing good for the people who loved fashion and related trend.

    ReplyDelete
  6. Hello. Thanks for the interesting article.
    Yes, Excel spreadsheets are a very cool program. But, I would say, very few use all its possibilities. This is more for various formulas, calculations, reports, serious tables.
    Many students use their usual text editor when performing their tasks ...

    ReplyDelete
  7. Great article! I found your website perfect for my needs. It contains wonderful and helpful posts. Keep up the good work!. Thank you for this wonderful Article! Look here!
    http://edu-answers.com

    ReplyDelete
  8. So why choose us when looking for professional assistance on writing successful medical school essays ? The answer is obvious. We are the team that conceals no information from our customers. You are always welcome to browse our reviews section to discover how satisfied our customers are. We are very proud of our good name and reputation and we are happy to see every single positive response and evaluation of our best practices. From the number of positive responses on our website you may see that we excel ourselves to meet the highest standards you require and we are extremely good at that.
    The fact that you are on cosmetic surgery essay topics right now means you have found a company of professional essay writers you can trust. A reputable team of highly educated and vastly experienced essayists, we will make sure that the piece of writing you purchase from us is the best one you can imagine. The writers from our agency have the necessary education to give you a very good argumentative essay on health care or an original research paper on health care because health care issues are among the most prioritized directions of our business.

    ReplyDelete
  9. With bulk texting you can achieve what most businesses are aiming for which is real time communication. This enables businesses to send out daily promotions in real time and ensure that they are read within minutes of delivery.

    ReplyDelete
  10. Udaipur escort service, Haldwani escort service, Ludhiana call girls Ludhiana escorts are educated along with experienced and they Have the propensity to catch the attention of consumers with their sex appeal and glorious bodily structure. That's to say they might be called intriguing, sexy, and alluring.
    Ludhiana Call Girls
    Dehradun Call Girls
    Jodhpur Call Girls
    Gurgaon Call Girls
    Noida Call Girls
    Chandigarh Call Girls

    ReplyDelete
  11. The infrastructure improvements allowed the company to expand its customer base, and business partners praised the solution's potential to optimize the ad campaign management process. Integrating well with the internal protocol, telemedicine platform development ​delivered quality work and communicated efficiently.

    ReplyDelete
  12. The projects continue to meet expectations and perform reliably. As a team, telemedicine platform development have been receptive to feedback and are willing to modify their process for satisfaction. They’re organized and quick to respond, making them an accommodating partner in the ongoing engagement.

    ReplyDelete
  13. Get the charms of lovemaking session through our Blessed messenger Escorts who are continuously prepared to be in a relationship with a individual like you who are amazingly upbeat in brief term one night stand instead of full long term relationship?

    Surat cheap rate call girls phone number
    low rate call girls in surat
    beautiful photo gallery escorts in surat
    beautiful escort in surat
    independent escort in surat

    ReplyDelete
  14. . Their arousing love chomps will make the take in your body parts will be difficult to be a long way from them. They are such excellent with the goal that you will never leave them. Their elusive body, their conditioned skins are best for eye mitigating. They are best for your body relieving and for the joy past your limits. AHMEDABAD Escorts are the world best escorts known for their magnificent positions, for the full delight to their customers and their enticing highlights.
    chhotaudaipur escorts
    dahod escorts
    daman escorts
    diu escorts
    dwarka escorts
    gandhinagar escorts

    ReplyDelete
  15. Escorts vadodara Call For Booking: Our vadodara Escorts Service Girls is an Independent High class Profile Society Female, Use For Fun enjoy, date dinner payable ...


    Click here : vadodara escorts service

    ReplyDelete
  16. Escort you will act the hero! This is a work for young ladies in Kolkata , which can stand to be delightful as well as cheerful. You will meet numerous new and intriguing individuals, spend oligarchs society, representatives and financial specialists energizing night will be held in exclusive hangouts, planned for VIPs ... You will be accessible to new and tremendous freedoms! All things considered, for the entirety of this and better paid! Occupations for Women - great!

    mountabu escorts
    dakshin-dinajpur escorts
    hooghly escorts
    howrah escorts
    jalpaiguri escorts

    ReplyDelete
  17. We are the best escort agency in Chandigarh. We know hygiene matters to our customers, so our escorts take good care of their hygiene. They always stay fresh and well maintained.

    Chandigarh Escorts
    Mohali Escorts
    Zirakpur Escorts
    Panchkula Escorts
    Amritsar Escorts
    Chandigarh Call Girls

    ReplyDelete
  18. If you are looking for some sexy friendship, a dinner meeting, or a social party with female escorts in agency, hence contact us freely. We are very well with the demands of our customers. All our workers and surat escort girls play a great role to make available the finest pleasure to our beloved clientele.

    surat escorts

    ReplyDelete
  19. . In our surat escort agency, we also have surat independent escorts who would like to be a honey baby for you. We make available celebrity escorts services to our important customers. Our most important goal is the customer pleasure. After getting trained on the guide line of escort services she will be the finest woman for meeting and perfect intercourse partners.

    escort in surat

    ReplyDelete
  20. We are the only escort supplier who have real women having know-how in offer sessions. Our girls are well educated and well affected. They know how to complete the erotic wants of our beloved customers.

    independent surat escorts

    ReplyDelete
  21. we and our independent female friends like to go with business men and young mens those like to get service with us you will get high class independent service female we all are professional for our service.
    real photo service escorts in buldana
    sexy female escorts in bid
    hotel escorts in aurangabad
    call girls service escorts in akola
    sexy female escorts in ahmadnagar

    ReplyDelete
  22. At the moment, if you are not personally connected to Latvia or anyone already there in a way which would entitle you to a residence permit (i.e. EU Citizenship, Family reunification, Family immigration, Latvian Ancestry etc.), then there still exists in Latvia two major ways to receive a residence permit: http://www.immigration-residency.eu/residence-permit-latvia/

    ReplyDelete
  23. This implies getting the sort of staggering modest Kolkata Escorts that most different offices can just fantasy about getting. Our exhaustive exploring measure guarantees that simply the best possibilities become obvious and once they do we make it our central goal to discover the genuinely splendid from that generally superb group. You can have confidence that when you get one of our young ladies, she will sparkle.
    mountabu escorts
    kolkata escorts
    kolkata-call-girls-whatsapp-number-escorts
    dakshin-dinajpur escorts
    hooghly escorts

    ReplyDelete
  24. “Interesting  information, thanks for making these contributions.
    Great  information Glad to find your article.!This blog is really very informative.
    Thanks for sharing it with us
    bangalore escorts service 
     bangalore call girls 
    bangalore russian escorts 
    bangalore cheap escorts 
    bangalore escorts 

    ReplyDelete
  25. https://alessandrobarbucci.blogspot.com/2012/09/rentree-en-plein-forme-cramp-de.html?showComment=1623157966677#c6912230240657986146

    ReplyDelete


  26. Do you want to get a full exotic service? Mumbai Escorts Hub here at your service. You can meet with our call girls to make your time memorable full of enjoyment. We offer all types of sex with great satisfaction. Our Independent call Girl gives real massage along with full service of deep throat, cim, cum, anal and much more basically you think our ladies and would love to serve you without rushing. It's our passion and choice to be an escort that's why our call girls love to satisfy our clients for recommendation and referral.

    Mumbai Escorts Services:
    Mumbai Escorts | Mumbai Escorts Service |

    Mumbai Escorts Categories:
    Mumbai Sex Aunty Number | Phone Sex Mumbai | Mumbai Call Girl Whatsapp Number | Mumbai Call Girls Near Me |

    ReplyDelete
  27. Those persons, who are doing jobs in office, they can improve their knowledge about excel and do their work in short time. I hope, they will get benefits from these tips. Dissertation editing services.

    ReplyDelete
  28. Their broad-spectrum method isn't solely potent and nice tasting (we’re particularly keen on the mint flavor), it’s fairly priced as well. If we had to choose one from the bunch, it’d most likely should be pineapple. Nothing says “tropical” fairly like its delicately balanced sweet-but-tart taste. We’d recommend getting the Fruit Pack if attainable for a real celebration.

    UltraSonic Keto
    One Shot Keto Pro
    Insulux Comentarios
    UltraSonic Keto Pills

    ReplyDelete
  29. Call girls in Dubai with Sexy Indian Girls for Dubai Escorts sex with CALL GIRLS NEAR ME or hot indian girls local call girl mobile number are available for romance with call girls in Dubai @ girls mobile number list at hot indian girls. We call girls in Dubai have beautiful call girls service

    Abu Dhabi Call Girls

    Abu Dhabi Escorts

    Dubai Call Girls

    Escort Service In Dubai

    Dubai Escort Service

    Dubai Escorts

    Dubai Call Girls

    Dubai Escorts Call Girls

    Sharjah Call Girls

    Vip Dubai Call Girls

    Ajman Escort Service

    Bangalore Call Girls

    Dubai Escorts

    Sharjah Escort Service

    Goa Escort Service

    Bur Dubai Call Girls

    Marina Call Girls

    Bur Dubai Escort Service

    Marina Escort Service

    Call Girls

    ReplyDelete
  30. You'll soon find out how Ketosis Diet Pills actually consumes your time. I'm thinking about setting up a private membership site as this respects Ketosis Diet Pills. I wonder how many Ketosis Diet Pills I have? It has irresistible appeal. Leave no stone unturned when it draws a parallel to Ketosis Diet Pills. There has been a high degree of this recently. I may want to give Ketosis Diet Pills a run for the money. This is a bit of information as to describing it with that. This is adequate.

    https://www.supplementz.org/keto-burn-am-reviews/

    https://www.supplementz.org/trim-life-keto/
    https://www.supplementz.org/exipure-reviews/
    https://www.supplementz.org/testotin-reviews/
    https://www.supplementz.org/nutri-acv-gummies/
    https://www.eunews24.com/sponsored/mens-miracle-health-reviews-male-enhancement-for-larger-penis-increased-libido/
    https://www.eunews24.com/sponsored/tvolve-gt-5-muscle-complex-testosterone-booster-reviews-legit-or-scam/
    https://www.comunidadnews.com/2021/11/10/mens-miracle-health-male-enhancement-worth-buying-or-scam-reviews-2021/

    ReplyDelete
  31. I really like the look of your site and its content. We callgirlschandigarh.com to wait for the latest from this site.

    ReplyDelete
  32. Для примера, клиенту требуется подтолкнуть группу или обсуждение в соц.сети, тогда подбираются пользователи, что должны вступать в группы и публиковать хорошие сообщения. Необходимо помнить, что, цена за элементарные задания не слишком большая, но и специальных квалификаций эта работа не потребует. С целью осуществления особых заданий требуется повозиться с недорогими проектами гидра сайт длительное время.

    ReplyDelete
  33. This was well planned as you'll know all as this regards to Outback Belly Burner in just a short amount of time. If some old hands are expecting a Outback Belly Burner, as a matter of fact it can occur. How do we do it? Where can aficionados bump into distinguished Outback Belly Burner advice?

    https://www.supplementz.org/outback-belly-burner/

    https://www.supplementz.org/pt-trim/
    https://www.supplementz.org/ultra-proven/
    https://www.supplementz.org/lean-time-keto/
    https://www.supplementz.org/optimal-max-keto/
    https://www.supplementz.org/keoni-cbd-gummies/
    https://www.supplementz.org/king-cobra-gummies/
    https://www.supplementz.org/minciball-forte-avis/

    ReplyDelete
  34. We have served numerous customers in our country, so we understand how to keep customer satisfied so that they can come back again and refer their friends too! Bangalore Naughty Girls

    ReplyDelete
  35. You can get my service as per your desire and feel high as like heaven. You will enjoy my amazing personality and feel free to share your inner feelings with me. Mumbai Sexy Girls

    ReplyDelete
  36. I every time spent my half an hour to read this weblog's posts daily along with a cup of coffee. 메이저사이트

    ReplyDelete
  37. Hello There. I discovered your blog using msn. That is a really well written article.
    I will be sure to bookmark it and come back to learn extra of your helpful information. Thanks for the post.
    I'll certainly comeback 온라인경마

    ReplyDelete
  38. Hey! I'd like to start a blog so I can easily share my own experience and feelings online. Please let me know if you have any kind of ideas or tips for new aspiring blog owners. 카지노

    ReplyDelete
  39. Woah! I'm really enjoying the template/theme of this website.
    It's simple, yet effective. A lot of times it's difficult to get that "perfect balance" between usability and visual appearance. 사설토토

    ReplyDelete
  40. I am providing it with my friends as they are at the same time looking out for this kind of illuminative blogs.
    december umrah packages

    ReplyDelete
  41. Do you want to know which email service in Australia is the best? Is it difficult for you to get emails in Australia? You no longer have to be concerned about this. You can use the Optus Email Service, which is a fantastic email service. Due to the fact that email is frequently ignored or filtered as spam. To solve this, Optus has established the Optus Tech Support Number Australia 1-800-431-401, which is managed by third-party firm Helpdesk Australia. If we have an Optus difficulty, we can call 1-800-431-401 Optus Tech Support Number Australia for immediate assistance.

    ReplyDelete
  42. This is a Great Article I Am enjoying When I Reading It.That is Really an Wonderfull Post

    ReplyDelete
  43. I love being in front of the camera. I mainly shoot fashion

    ReplyDelete
  44. female to male spa at home near me is a type of bodywork that involves the manipulation of soft tissues, such as muscles, connective tissue, tendons, and ligaments.

    ReplyDelete
  45. very useful content written thanks for sharing with us.

    ReplyDelete
  46. Discover a realm of tranquility and relaxation that can revitalize both your body and mind. In this comprehensive guide, we unveil the secrets to a remarkable spa experience, full night service near me

    ReplyDelete
  47. Once the range of motion of those muscles returns, then you can build up strength by doing pelvic floor exercises.

    ReplyDelete
  48. and you can use them close to bedtime or throughout your day.

    ReplyDelete
  49. . These feelings occur because your muscles are getting a workout from above, and the table supports you from below.

    ReplyDelete
  50. You can stimulate your man’s prostate through his anus or perineum. Check out this information to learn more.

    ReplyDelete
  51. female to male body massage centres near me a ancient practice intertwined with modern wellness, employs heated volcanic rocks strategically placed on key points of the body.

    ReplyDelete
  52. The Steric touch, marked by meticulous attention to detail, transforms your session into a personalized journey of massage parlour near me

    ReplyDelete
  53. Thank you for providing such a beautiful piece of information.

    ReplyDelete
  54. very interesting, good job and thanks for sharing such a good blog.

    ReplyDelete
  55. there is no need to worry if your body is suffering from headache, come to massage in chennai center just once and feel how relaxed you are, our hot massage girl therapist will take all your pain

    ReplyDelete
  56. Spa treatments provide an opportunity to relax, unwind, and escape from daily stressors, this spa center provide best experienced therapist, so come and enjoy with our therapist

    ReplyDelete
  57. Invaluable insights! Learning measures of variation in Excel is a game-changer. As someone focused on Umrah Packages, mastering Excel boosts productivity. Can't wait to use these techniques. Thanks for the helpful guide!

    ReplyDelete
  58. Chiropractic card processing solutions are tailored to meet the unique needs of chiropractic practices, offering features such as recurring billing for patients on payment plans, online payment portals for patient convenience, and integration with popular chiropractic software systems. With secure payment processing, chiropractors can ensure that patient information is protected and transactions are processed efficiently.

    By implementing Chiropractic Card Processing services, chiropractors can focus on providing quality care to their patients while leaving the payment processing to a reliable and efficient system.

    ReplyDelete
  59. If you have failed in many attempts to enhance your beauty, then do not hesitate to come to our female to male massage centres in chennai and get the services of our spa experts to achieve a heavenly beauty.

    ReplyDelete
  60. Awesome post. I am a regular visitor of your blog and appreciate you taking the time to maintain the nice site. Having read this I believed it was extremely informative. I appreciate you finding the time and energy to put this information together.

    ReplyDelete